A structure theory for signed graphs with fixed smallest eigenvalue

A structure theory for signed graphs with fixed smallest eigen v alue Jac k H. Koolen a,b , Jing-Y uan Liu a , Qianqian Y ang c,d , and Meng-Y ue Cao ∗ a a School of Mathematical Sciences, Universit y of Science and T echnology of China, Hefei, 230026, P eople’s Republic of China b CAS W u W en-Tsun Key Laboratory of Mathematics, Universit y of Science and T ec hnology of China, Hefei, 230026, People’s Republic of China c Department of Mathematics, Shanghai Univ ersity , Shanghai 200444, P eople’s Republic of China d Newtouch Cen ter for Mathematics of Shanghai Universit y , Shanghai 200444, People’s Republic of China Abstract In this pap er, we will giv e a structure theory for signed graphs with fixed smallest eigenv alue and in vestigate signed graphs with smallest eigenv alue greater than − 1 − √ 2 . Given a real num ber λ ≤ − 1, w e sho w that the following hold for each signed graph ( G, σ ) with smallest eigen v alue at least λ and large minim um v alency: (i) there exist dense induced subgraphs N 1 , . . . , N r in ( G, σ ) such that eac h vertex lies in at most ⌊− λ ⌋ N i ’s and almost all edges of ( G, σ ) lie in at least one of the N i ’s; (ii) if λ > − 1 − √ 2 , then ( G, σ ) has smallest eigenv alue at least − 2 and ( G, σ ) is 1-in tegrable. Keyw ords : Signed graph; Smallest eigen v alue; Structure theory; Hoffman signed graph MSC2020 : 05C50, 05C22, 52C35, 05C62, 05C75 1 In tro duction All graphs mentioned in this pap er are finite, undirected, simple and non-empt y . A signe d gr aph ( G, σ ) is a pair of a graph G = ( V ( G ) , E ( G )) and a signing σ : E ( G ) → { + , −} . W e call G the underlying gr aph of ( G, σ ). W e sa y ( G, σ ) is isomorphic to ( H , τ ) if there exists an isomorphism ψ : G → H suc h that σ ( { x, y } ) = τ ( { ψ ( x ) , ψ ( y ) } ) for all { x, y } ∈ E ( G ). The valency of a vertex x in ( G, σ ) is the cardinality of the set { y ∈ V ( G ) | { x, y } ∈ E ( G ) } . The p ositive (resp. ne gative ) gr aph of ( G, σ ) is the unsigned graph ( G, σ ) + (resp. ( G, σ ) − ) with v ertex set V ( G ) such that the edge set E (( G, σ ) + ) = {{ x, y } ∈ E ( G ) | σ ( { x, y } ) = + } (resp. E (( G, σ ) − ) = {{ x, y } ∈ E ( G ) | σ ( { x, y } ) = −} ). W e mean by a sub gr aph of a signed graph ( G, σ ) on vertex set U ⊆ V ( G ), the signed graph ( H , σ H ) with vertex set U suc h that E ( H ) ⊆  V ( H ) 2  ∩ E ( G ) and σ H = σ | E ( H ) . If E ( H ) =  V ( H ) 2  ∩ E ( G ), w e say ( H , σ H ) is an induc e d sub gr aph of ( G, σ ). Let U ⊆ V ( G ). W e sa y the signed graph ( G, τ ) is obtained from ( G, σ ) by switching with respect to U , if the signing τ satisfies the following: for any edge { x, y } of G , τ ( { x, y } ) = σ ( { x, y } ) if x, y are b oth in U or b oth in V ( G ) \ U , and τ ( { x, y } )  = σ ( { x, y } ) otherwise. If ( G, τ ) can b e obtained from ( G, σ ) b y switching, w e sa y that ( G, τ ) and ( G, σ ) are switching e quivalent . Given a signed graph ( H , τ ), we sa y the signed graph ( G, σ ) is ( H , τ ) -switching-fr e e , if ( G, σ ) contains no induced subgraphs switching equiv alen t to ( H, τ ). As in the case of unsigned graphs, we define the adjac ency matrix A = A ( G, σ ) of ( G, σ ) to b e the symmetric matrix indexed b y V ( G ) such that A xy =    1 , if { x, y } ∈ E ( G ) and σ ( { x, y } ) = + , − 1 , if { x, y } ∈ E ( G ) and σ ( { x, y } ) = − , 0 , otherwise . The eigenvalues of ( G, σ ) are the eigenv alues of the adjacency matrix A . ∗ Corresponding author E-mail addresses: koolen@ustc.edu.cn (J.H. Ko olen), liujingyuan@mail.ustc.edu.cn (J.-Y. Liu), qqyang@shu.edu.cn (Q. Y ang), caomengyue@ustc.edu.cn (M.-Y. Cao) 1 In this pap er, we study signed graphs with a fixed smallest eigenv alue. W e will see that several results for the unsigned case can be generalized to signed graphs. In order to state the results, we need to in tro duce sev eral notations. F or a graph G , we write ( G, +) the signed graph ( G, σ ) such that σ ( { x, y } ) = + for all edges { x, y } ∈ E ( G ), and ( G, − ) the signed graph ( G, σ ) such that σ ( { x, y } ) = − for all edges { x, y } ∈ E ( G ). With K n w e will denote the complete graph on n v ertices. Let t b e a positive in teger. W e define e K ( ε ) 2 t for ε ∈ { 0 , −} as the signed graph ( G, σ ) on vertex set { 0 , 1 , . . . , 2 t } such that { i, j } ∈ E ( G ) and σ ( { i , j } ) =    + , if 1 ≤ i, j ≤ 2 t, + , if i = 0 , 1 ≤ j ≤ t, − , if i = 0 , t < j ≤ 2 t and ε = − . Later it will be clear why we use this notation. F or examples of e K ( ε ) 2 t for t = 2, see Fig. 1 . f K 4 (0) f K 4 ( − ) Figure 1: The signed graphs of f K 4 (0) and f K 4 ( − ) (Edges with sign + are repr esente d by solid se gments and e dges with sign − ar e r epr esented by dashe d se gments.) A t -plex is an unsigned graph such that each vertex has at most t − 1 non-neighbors. Now w e are able to state our first main result. Theorem 1.1. L et ( G, σ ) b e a signe d gr aph with smal lest eigenvalue λ min ( G, σ ) . The fol lowing hold. (i) F or any r e al numb er λ ≤ − 1 , ther e exists a p ositive inte ger t = t ( λ ) , such that if λ min ( G, σ ) ≥ λ , then ( G, σ ) is { e K (0) 2 t , e K ( − ) 2 t , ( K t +1 , − ) , ( K 1 ,t , +) } -switching-fr e e. (ii) F or any p ositive inte ger t , ther e exists a non-p ositive r e al numb er λ = λ ( t ) , such that if ( G, σ ) is { e K (0) 2 t , e K ( − ) 2 t , ( K t +1 , − ) , ( K 1 ,t , +) } -switching-fr e e, then λ min ( G, σ ) ≥ λ . Remark 1.2. (i) Note that for any signe d gr aph in the switching class of e K (0) 2 t , e K ( − ) 2 t , ( K t +1 , − ) or ( K 1 ,t , +) , its smal lest eigenvalue go es to −∞ when t go es to + ∞ . This implies that (i) fol lows fr om the interlacing the or em [ 7 , The or em 9 . 9 . 1 ]. We wil l give a pr o of of (ii) in Se ction 5 . (ii) Hoffman [ 10 ] showe d that Thr or em 1.1 (i) also holds if we r eplac e the { e K (0) 2 t , e K ( − ) 2 t , ( K t +1 , − ) , ( K 1 ,t , +) } - switching-fr e e c ondition by ten p articular signe d gr aphs. W e also show the following refinemen t of Theorem 1.1 . Theorem 1.3. L et λ ≤ − 1 b e a r e al numb er. Ther e exists a p ositive inte ger d λ such that if ( G, σ ) is a signe d gr aph with smal lest eigenvalue at le ast λ and minimum valency at le ast d λ , then ther e exists a set of induc e d sub gr aphs N 1 , N 2 , . . . , N r of ( G, σ ) , wher e r is a p ositive inte ger, satisfying the fol lowing c onditions. (i) Each vertex of ( G, σ ) lies in at le ast one and at most ⌊− λ ⌋ N i ’s. (ii) The induc e d sub gr aph N i is switching e quivalent to a signe d gr aph whose p ositive gr aph is a ( ⌊ λ 2 + 2 λ + 2 ⌋ ) - plex, for i = 1 , 2 , . . . , r . (iii) The interse ction V ( N i ) ∩ V ( N j ) c ontains at most 4 ⌊− λ ⌋ − 4 vertic es for 1 ≤ i < j ≤ r . 2 (iv) the sub gr aph ( G ′ , σ ′ ) has maximum valency at most d λ − 1 , wher e G ′ = ( V ( G ) , E ( G ) \ S r i =1 E ( N i )) and σ ′ = σ | E ( G ′ ) . Remark 1.4. Note that for the unsigne d c ase, The or em 1.1 was shown by Hoffman [ 8 ] and The or em 1.3 was shown by Kim et al. [ 12 ]. Ga vrilyuk et al. [ 5 ] show ed the following result. Theorem 1.5 ([ 5 , Theorem 1 . 2]) . Ther e exists an inte ger value d function f define d on the half-op en interval ( − 2 , − 1] such that, for e ach λ ∈ ( − 2 , − 1] , if a c onne cte d signe d gr aph ( G, σ ) has smal lest eigenvalue at le ast λ and minimum valency at le ast f ( λ ) , then ( G, σ ) is switching e quivalent to a c omplete gr aph (with e ach e dge signe d + ) and henc e λ min ( G, σ ) = − 1 . W e extend this result to the in terv al ( − 1 − √ 2 , − 1]. T o state the result, w e need to define s -in tegrable signed graphs analogue to the same notion for in tegral lattice where s is a p ositiv e in teger. F or more details of lattices, see Subsection 4.2 . Let s b e a positive in teger. A signed graph ( G, σ ) with smallest eigenv alue λ min is s -inte gr able , if there exists an integer-v alued matrix N suc h that s ( A + ⌈− λ min ⌉ I ) = N T N , where A is the adjacency matrix of G . Note that ( G, σ ) is s -in tegrable if and only if the in tegral lattice generated by the columns of 1 √ s N is s -in tegrable. Our result is: Theorem 1.6. L et λ b e a r e al numb er in ( − 1 − √ 2 , − 1] . Ther e exists a p ositive inte ger d ′ λ such that if a c onne cte d signe d gr aph ( G, σ ) has smal lest eigenvalue λ min ( G, σ ) ≥ λ and minimum valency at le ast d ′ λ , then λ min ( G, σ ) ≥ − 2 and ( G, σ ) is 1 -inte gr able. Remark 1.7. (i) Note that The or em 1.6 states that for − 1 − √ 2 < λ ≤ − 2 , if a signe d gr aph ( G, σ ) has smal lest eigenvalue λ min ( G, σ ) ≥ λ and lar ge enough minimum valency, then its adjac ency matrix A ( G, σ ) satisfies that A ( G, σ ) + 2 I = N T N for some { 0 , ± 1 } -matrix N . (ii) The or em 1.5 and The or em 1.6 wer e shown by Hoffman [ 9 ] for the unsigne d c ase, as a gener alize d line gr aph is an unsigne d gr aph with adjac ency matrix A such that A + 2 I = N T N for some { 0 , ± 1 } -matrix N . (iii) F or λ ≥ − 2 , Belar do et al. impr ove d The or em 1.6 as fol lows: Theorem 1.8 ([ 3 , Theorem 3 . 13]) . Each c onne cte d signe d gr aph ( G, σ ) with smal lest eigenvalue at le ast − 2 is 2 -inte gr able. Mor e over, if ( G, σ ) has at le ast 121 vertic es, then ( G, σ ) is 1 -inte gr able. This r esult was pr ob ably known by Hoffman, as he studie d signe d gr aphs in [ 10 ]. 1.1 Line systems, F 3 -w eigh ted complete graphs and signed graphs In this subsection, w e lo ok at line systems in the Euclidean space R d . Let L b e a line system. Represen t the lines in L b y unit vectors u 1 , . . . u n and denote by Gr the Gram matrix of these v ectors. If L is a set of equiangular lines, which means that there exists an real num b er α  = 0 such that ( u i , u j ) ∈ {± α } for 1 ≤ i  = j ≤ n , then the matrix 1 α ( Gr − I ) is a symmetric matrix with only ± 1 off the diagonal. This is the so-called Seidel matrix, whic h was first in tro duced b y V an Lint and Seidel in [ 15 ]. If the lines of L form tw o different angles, arccos α and 90 ◦ , then the matrix 1 α ( Gr − I ) is a symmetric matrix with 0 on the diagonal and 0 or ± 1 off the diagonal, and we call it the gener alize d Seidel matrix of { u 1 , . . . u n } . Problem 1.9. Given a p ositive r e al numb er α and a p ositive inte ger d , what is the maximal numb er n such that ther e exists a line system of n lines in the Euclide an sp ac e R d with angles arccos α and 90 ◦ ? 3 It is known that for a p ositiv e in teger d , w e ha ve n ≤  d +3 3  , as the set of vectors { u 1 , . . . u n } forms a spherical Euclidean 3-distance set (see [ 4 ] and [ 2 ]). In [ 6 ], an upp er bound  d +1 2  for the cardinalit y of a spherical Euclidean 2-distance set is prov en for most d . W e guess that the b ound for the cardinality of a spherical Euclidean 3-distance set may be possible to b e impro ved to  d +2 3  , and if so this will giv e an upper b ound of n in Problem 1.9 . In order to solv e Problem 1.9 , it is essen tial to study { 0 , ± 1 } -matrices with fixed smallest eigen v alue. Let S b e a symmetric matrix with 0 on the diagonal and 0 or ± 1 off the diagonal, and { u 1 , . . . , u n } b e a set of vectors with Gram matrix Gr = αS + I . It is natural to regard the normalized inner products 1 α ( u i , u j ) as the weigh t of the edge { i, j } in K n , b y which we obtain a weigh ted complete graph K n with w eights { 0 , ± 1 } . It is reasonable to consider { 0 , ± 1 } as the field F 3 with 3 elements, hence w e can consider S as the adjacency matrix of a F 3 -w eighted complete graph. Another wa y to lo ok at S , see also [ 13 , 1 ] for similar discussions, is to regard S as the adjacency matrix of a signed graph ( G, σ ), where the v ertex set is { 1 , . . . , n } , the edge set is {{ i, j } | ( u i , u j )  = 0 } , and the signing is σ ( { i, j } ) = + (resp. σ ( { i, j } ) = − ) if 1 α ( u i , u j ) = +1 (resp. 1 α ( u i , u j ) = − 1). In the first viewp oin t, we consider 0 and ± 1 all the same, whereas in the second viewp oin t, 0 pla ys a different role from ± 1. In this pap er, we will mainly use the language of signed graphs, except in Section 2 . This pap er is organized as follo ws. In Section 2 , w e will define F 3 -w eighted complete graphs and state some results; in Section 3 , w e will in tro duce more definitions of signed graphs, and translate the results for F 3 -w eighted complete graphs to signed graphs; in Section 4 , we will in tro duce Hoffman signed graphs and asso ciated Hoffman signed graphs; in Section 5 , w e will pro ve Theorem 1.1 (ii) and Theorem 1.3 , and in Section 6 , we will fo cus on signed graphs with smallest eigen v alue greater than − 1 − √ 2 and give a pro of of Theorem 1.6 . 2 F 3 -w eigh ted complete graphs 2.1 Definitions of F 3 -w eigh ted complete graphs In this subsection, we in tro duce some definitions and notations of F 3 -w eighted complete graphs. Recall that F 3 is the field with elemen ts { 0 , ± 1 } . A F 3 -weighte d c omplete gr aph ( K, w ) is a pair of a complete graph K and a w eight w :  V ( K ) 2  → { 0 , ± 1 } . W e say ( K, w ) is isomorphic to ( K ′ , w ′ ) if there exists an isomorphism ψ : V ( K ) → V ( K ′ ) such that w ( { x, y } ) = w ′ ( { ψ ( x ) , ψ ( y ) } ) for all { x, y } ∈  V ( K ) 2  . W e mean by the induc e d sub gr aph of the F 3 -w eighted complete graph ( K, w ) on vertex set U ⊆ V ( K ), the F 3 -w eighted complete graph ( K 1 , w K 1 ) such that V ( K 1 ) = U , E ( K 1 ) =  U 2  and w K 1 = w | E ( K 1 ) . W e denote b y ( K, w ) U the induced subgraph of ( K, w ) on U ⊆ V ( K ). Let ε ∈ { 0 , ± 1 } . F or an y pair of vertices x, y of K , we sa y x and y are ( ε ) -neighb ors , when w ( { x, y } ) = ε . Let x b e a v ertex of the F 3 -w eighted complete graph ( K, w ). The ( ε ) -neighb orho o d N ( ε ) ( x ) of x in ( K, w ) is the set of ( ε )-neigh b ors of x . F or a positive integer t and a vertex subset U ⊆ V ( K ), the t - ( ε ) -neighb orho o d of U in ( K, w ), denoted as N ( ε ) ( t ) ( U ), is the vertex set { x ∈ V ( K ) | x has at least t ( ε )-neigh b ors in U } . W e call the F 3 -w eighted complete graph ( K, w ) an ( ε ) -clique , and denoted b y ( K, ε ), if w ( { x, y } ) = ε for all { x, y } ∈  V ( K ) 2  . F or the F 3 -w eighted complete graph ( K, w ), we define the adjac ency matrix A = A ( K, w ) to be the symmetric matrix indexed b y V ( K ) such that A xy =  w ( { x, y } ) , if x  = y , 0 , if x = y . The eigenvalues of ( K, w ) are the eigenv alues of its adjacency matrix. W e sa y the F 3 -w eighted complete graph ( K, w ′ ) is o b tained from ( K, w ) b y switching with resp ect to a vertex subset U ⊆ V ( K ), if w ′ satisfies the following: for an y edge { x, y } of K , w ′ ( { x, y } ) = w ( { x, y } ) if x, y are b oth in U or b oth in V ( K ) \ U , and w ′ ( { x, y } ) = − w ( { x, y } ) otherwise. If ( K, w ′ ) can b e obtained 4 from ( K, w ) by switc hing, w e say that ( K, w ) and ( K, w ′ ) are switching e quivalent . Note that eac h switching equiv alen t F 3 -w eighted complete graphs are cosp ectral. Given a F 3 -w eighted complete graph ( H , w H ), we sa y the F 3 -w eighted complete graph ( K, w ) is ( H , w H ) -switching-fr e e , if ( K, w ) contains no induced subgraphs that are switching equiv alen t to ( H , w H ). By considering the non-edges in signed graphs as (0)-edges in F 3 -w eighted complete graphs, a signed graph can b e regarded as a F 3 -w eighted complete graph. W e denote b y e K (0) 2 m , e K ( − 1) 2 m and ( K 1 ,t , +1) the corresp onding F 3 -w eighted complete graphs of signed graphs e K (0) 2 m , e K ( − ) 2 m and ( K 1 ,t , +), resp ectiv ely . Now it is clear why w e use the notations for signed graphs e K ( ε ) 2 m for ε ∈ { 0 , −} . By considering the w eights as colors, the Ramsey theorem in three colors case can b e applied to F 3 -w eighted complete graph. Theorem 2.1 ([ 14 ]) . L et m, s and t b e thre e p ositive inte gers. Ther e exists a minimum p ositive inte ger R ( m, s, t ) such that for any F 3 -weighte d c omplete gr aph with at le ast R ( m, s, t ) vertic es, it c ontains a (+1) - clique ( K m , +1) , a ( − 1) -clique ( K s , − 1) , or a (0) -clique ( K t , 0) . 2.2 Results on F 3 -w eigh ted complete graphs In this subsection, we sho w some prop erties on { e K (0) 2 m , e K ( − 1) 2 m } -switc hing-free F 3 -w eighted complete graphs for integer m ≥ 2. Lemma 2.2. L et m ≥ 2 , n ≥ 3 m − 2 b e inte gers and ( K, w ) b e a { e K (0) 2 m , e K ( − 1) 2 m } -switching-fr e e F 3 -weighte d c omplete gr aph. L et C b e a (+1) -clique of ( K, w ) with at le ast n vertic es. F or any vertex x ∈ V ( K ) , one of the fol lowing holds. (i) x ∈ N (+1) ( m ) ( V ( C )) , and x has at most m − 1 ( − 1) -neighb ors and at most m − 1 (0) -neighb ors in C . (ii) x ∈ N ( − 1) ( m ) ( V ( C )) , and x has at most m − 1 (+1) -neighb ors and at most m − 1 (0) -neighb ors in C . (iii) x ∈ N (0) ( m ) ( V ( C )) , and x has at most m − 1 (+1) -neighb ors and at most m − 1 ( − 1) -neighb ors in C . Pr o of. F or any vertex x ∈ V ( K ), at least one of the three sets N (+1) ( x ) ∩ V ( C ), N ( − 1) ( x ) ∩ V ( C ) and N (0) ( x ) ∩ V ( C ) contains at least m v ertices, as n ≥ 3 m − 2. Without loss of generality , we ma y assume that | N (+1) ( x ) ∩ V ( C ) | ≥ m , then x is con tained in the set N (+1) ( m ) ( V ( C )). Since ( K, w ) is { e K (0) 2 m , e K ( − 1) 2 m } -switc hing- free, the vertex x has at most m − 1 ( − 1)-neigh b ors and at most m − 1 (0)-neighbors in C . This completes the pro of. Lemma 2.3. L et m ≥ 2 , n ≥ 2( m 2 + m ) b e inte gers and ( K, w ) b e a { e K (0) 2 m , e K ( − 1) 2 m } -switching-fr e e F 3 -weighte d c omplete gr aph. L et C b e a (+1) -clique of ( K, w ) with at le ast n vertic es. L et ε ∈ { 0 , ± 1 } , ε ′ ∈ {± 1 } . F or any vertex x ∈ V ( K ) , if ther e exists a (+1) -clique D with m vertic es such that V ( D ) ⊆ N ( ε ) ( x ) ∩ N ( ε ′ ) ( m ) ( V ( C )) , then x ∈ N ( ε · ε ′ ) ( m ) ( V ( C )) . Pr o of. First, w e assume that ε ′ = +1, then V ( D ) ⊆ N ( ε ) ( x ) ∩ N (+1) ( m ) ( V ( C )). Since V ( D ) ⊆ N (+1) ( m ) ( V ( C )), eac h v ertex of D has at most 2 m − 2 non-(+1)-neighbors in C , by Lemma 2.2 . Thus, C con tains at least n − (2 m − 2) m ≥ 2( m 2 + m ) − 2 m 2 + 2 m = 4 m v ertices which are (+1)-neigh b ors of each vertex in D . This implies that | V ( D ) ∪ ( V ( C ) ∩ N (+1) ( m ) ( V ( D ))) | ≥ | V ( C ) ∩ N (+1) ( m ) ( V ( D )) | ≥ 4 m . Note that the subgraph induced on V ( D ) ∪ ( V ( C ) ∩ N (+1) ( m ) ( V ( D ))) is a (+1)-clique. As V ( D ) ⊆ N ( ε ) ( x ), the vertex x has at least 4 m − (2 m − 2) ≥ 2 m + 2 ( ε )-neigh b ors in V ( D ) ∪ ( V ( C ) ∩ N (+1) ( m ) ( V ( D ))), by Lemma 2.2 . Hence, the v ertex x has at least 2 m + 2 − m > m ( ε )-neighbors in C , in other w ords, the v ertex x ∈ N ( ε ) ( m ) ( V ( C )). No w w e assume that ε ′ = − 1, then V ( D ) ⊆ N ( ε ) ( x ) ∩ N ( − 1) ( m ) ( V ( C )). Since V ( D ) ⊆ N ( − 1) ( m ) ( V ( C )), eac h v ertex of D has at most 2 m − 2 non-( − 1)-neigh b ors in C , by Lemma 2.2 . Thus, C con tains at least 5 n − (2 m − 2) m ≥ 2( m 2 + m ) − 2 m 2 + 2 m = 4 m v ertices which are ( − 1)-neigh b ors of each vertex in D . W e claim that x has at least m ( − ε )-neigh b ors in C . Supp ose that x ∈ N ( − ε ) ( m ) ( V ( C )), then x has at most m − 1 ( − ε )-neigh b ors in C , by Lemma 2.2 . Thus, the vertex x has at least 4 m − ( m − 1) ≥ 3 m + 1 non-( − ε )-neigh b ors in V ( C ) ∩ N ( − 1) ( m ) ( V ( D )). By Lemma 2.2 , there exists η ∈ { 0 , ± 1 } , η  = − ε , such that there are at least m ( η )-neigh b ors of x in V ( C ) ∩ N ( − 1) ( m ) ( V ( D )). Let U b e a v ertex set of cardinalit y m suc h that U ⊆ V ( C ) ∩ N ( − 1) ( m ) ( V ( D )) ∩ N ( η ) ( x ). Let ( H , w H ) denote the subgraph of ( K, w ) induced on { x } ∪ U ∪ V ( D ), then ( H , w H ) is switching equiv alen t to e K (0) 2 m or e K ( − 1) 2 m , which is a con tradiction. Therefore, we obtain that x ∈ N ( − ε ) ( m ) ( V ( C )). This completes the proof. Let m ≥ 2 , n ≥ 2( m 2 + m ) b e integers and ( K, w ) b e a { e K (0) 2 m , e K ( − 1) 2 m } -switc hing-free F 3 -w eighted complete graph. Let C ( n ) denote the set { C | C is a maximal (+1)-clique of ( K, w ) with at least n v ertices } . Let C, C ′ ∈ C ( n ), the follo wing lemma sho ws the relation betw een the neighborho ods of C and C ′ . Lemma 2.4. L et m ≥ 2 , n ≥ 2( m 2 + m ) b e inte gers and ( K, w ) b e a { e K (0) 2 m , e K ( − 1) 2 m } -switching-fr e e F 3 -weighte d c omplete gr aph. F or any C , C ′ ∈ C ( n ) , one of the fol lowing holds. (i) Ther e exists an ε ∈ {± 1 } , such that | V ( C ′ ) ∩ N ( ε ) ( m ) ( V ( C )) | ≥ m , and N (+1) ( m ) ( V ( C ′ )) = N ( ε ) ( m ) ( V ( C )) , N ( − 1) ( m ) ( V ( C ′ )) = N ( − ε ) ( m ) ( V ( C )) . (ii) | V ( C ′ ) ∩ N (+1) ( m ) ( V ( C )) | ≤ m − 1 , | V ( C ′ ) ∩ N ( − 1) ( m ) ( V ( C )) | ≤ m − 1 , and at most 2 m − 2 vertic es of C ′ ar e not c ontaine d in N (0) ( m ) ( V ( C )) . Pr o of. By Lemma 2.2 , for an y v ertex of ( K, w ), it is contained in exactly one of the three vertex subsets N (+1) ( m ) ( V ( C )), N ( − 1) ( m ) ( V ( C )) and N (0) ( m ) ( V ( C )). If | V ( C ′ ) ∩ N (+1) ( m ) ( V ( C )) | ≤ m − 1 and | V ( C ′ ) ∩ N ( − 1) ( m ) ( V ( C )) | ≤ m − 1, then (ii) holds. No w we ma y assume that | V ( C ′ ) ∩ N ( ε ) ( m ) ( V ( C )) | ≥ m for some ε ∈ {± 1 } . W e claim that under this condition, N ( ε ′ ) ( m ) ( V ( C ′ )) ⊆ N ( ε · ε ′ ) ( m ) ( V ( C )) holds for an y ε ′ ∈ {± 1 } . T ake m v ertices from V ( C ′ ) ∩ N ( ε ) ( m ) ( V ( C )), then these vertices induce a (+1)-clique, denoted by D . Note that V ( D ) ⊆ N (+1) ( x ) for any vertex x ∈ V ( C ′ ) \ V ( D ). By Lemma 2.3 , w e obtain that x ∈ N ( ε ) ( m ) ( V ( C )). Thus, V ( C ′ ) ⊆ N ( ε ) ( m ) ( V ( C )). F or an y v ertex y ∈ N ( ε ′ ) ( m ) ( V ( C ′ )), it has at least m ( ε ′ )-neigh b ors in C ′ . This implies that there exists a ( K m , +1), in N ( ε ′ ) ( y ) ∩ V ( C ′ ) ⊆ N ( ε ′ ) ( y ) ∩ N ( ε ) ( m ) ( V ( C )). By Lemma 2.3 , w e obtain that y ∈ N ( ε · ε ′ ) ( m ) ( V ( C )). Hence, N ( ε ′ ) ( m ) ( V ( C ′ )) ⊆ N ( ε · ε ′ ) ( m ) ( V ( C )). T o complete the pro of, it suffices to show that N ( ε ′ ) ( m ) ( V ( C )) ⊆ N ( ε · ε ′ ) ( m ) ( V ( C ′ )) for an y ε ′ ∈ {± 1 } . This follo ws once we prov e that | V ( C ) ∩ N ( ε ) ( m ) ( V ( C ′ )) | ≥ m , by the claim in the previous paragraph. Since V ( D ) ⊆ N ( ε ) ( m ) ( V ( C )), each v ertex of D has at most 2 m − 2 non-( ε )-neigh b ors in C . By Lemma 2.2 , there exist at least n − (2 m − 2) m ≥ 2( m 2 + m ) − 2 m 2 + 2 m = 4 m v ertices in V ( C ) ∩ N ( ε ) ( m ) ( V ( D )). T ake m v ertices from this set, then these vertices induce a (+1)-clique, denoted by D ′ . Note that V ( D ′ ) is also con tained in N ( ε ) ( m ) ( V ( C ′ )), as V ( D ) ⊆ V ( C ′ ). Thus, | V ( C ) ∩ N ( ε ) ( m ) ( V ( C ′ )) | ≥ m . This completes the proof. Lemma 2.5. L et m ≥ 2 , s ≥ 3 , t ≥ 2 , n ≥ (2 m − 1) t + 1 b e inte gers. Ther e exists an inte ger κ 0 = κ 0 ( m, s, t ) such that for any { ( K s , − 1) , ( K 1 ,t , +1) } -switching-fr e e F 3 -weighte d c omplete gr aph ( K, w ) and any C ∈ C ( n ) , ε ∈ {± 1 } , if U ⊆ N ( ε ) ( m ) ( V ( C )) is a vertex subset with at le ast κ 0 vertic es, then the sub gr aph induc e d on U c ontains a (+1) -clique ( K m , +1) . Pr o of. Let κ 0 = κ 0 ( m, s, t ) := R ( m, s, t ), as defined in Theorem 2.1 . Let ( H , w H ) denote the subgraph induced on U . It follo ws from Lemma 2.2 that eac h vertex of ( H , w H ) has at most 2 m − 2 non-( ε )-neigh b ors 6 in C . By Theorem 2.1 , ( H , w H ) contains a ( K m , +1), ( K s , − 1), or a ( K t , 0). If ( H , w H ) contains a ( K t , 0), then there exists a vertex of C adjacen t to each vertex of this ( K t , 0), as n − (2 m − 2) t − t ≥ 1. This con tradicts that ( K, w ) is ( K 1 ,t , +1)-switc hing-free. Since ( K, w ) is ( K s , − 1)-switc hing-free, we obtain that ( H , w H ) contains a ( K m , +1). This completes the pro of. Lemma 2.6. L et m ≥ 2 , s ≥ 3 , t ≥ 2 , n ≥ max { 2( m 2 + m ) , (2 m − 1) t + 1 } b e inte gers. Ther e exists a p ositive inte ger κ 1 = κ 1 ( m, s, t ) such that for any { e K (0) 2 m , e K ( − 1) 2 m , ( K s , − 1) , ( K 1 ,t , +1) } -switching-fr e e F 3 -weighte d c omplete gr aph ( K , w ) , the fol lowing hold. (i) F or any x ∈ V ( K ) , C ∈ C ( n ) . (a) If x ∈ N (+1) ( m ) ( V ( C )) , then | N (+1) ( m ) ( V ( C )) ∩ N (0) ( x ) | ≤ κ 1 , | N (+1) ( m ) ( V ( C )) ∩ N ( − 1) ( x ) | ≤ κ 1 , | N ( − 1) ( m ) ( V ( C )) ∩ N (0) ( x ) | ≤ κ 1 , | N ( − 1) ( m ) ( V ( C )) ∩ N (+1) ( x ) | ≤ κ 1 . (b) If x ∈ N ( − 1) ( m ) ( V ( C )) , then | N (+1) ( m ) ( V ( C )) ∩ N (0) ( x ) | ≤ κ 1 , | N (+1) ( m ) ( V ( C )) ∩ N (+1) ( x ) | ≤ κ 1 , | N ( − 1) ( m ) ( V ( C )) ∩ N (0) ( x ) | ≤ κ 1 , | N ( − 1) ( m ) ( V ( C )) ∩ N ( − 1) ( x ) | ≤ κ 1 . (ii) F or any C 1 , C 2 ∈ C ( n ) , if N (+1) ( m ) ( V ( C 1 )) ∪ N ( − 1) ( m ) ( V ( C 1 ))  = N (+1) ( m ) ( V ( C 2 )) ∪ N ( − 1) ( m ) ( V ( C 2 )) , then | N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )) | ≤ κ 1 for any ε 1 , ε 2 ∈ {± 1 } . Pr o of. Let κ 1 = κ 1 ( m, s, t ) := κ 0 ( m, s, t ) − 1, as defined in Lemma 2.5 . (i) Suppose that x ∈ N ( ε 0 ) ( m ) ( V ( C )), where ε 0 ∈ {± 1 } . Let ε ∈ {± 1 } and ε ′ ∈ { 0 , − ε 0 · ε | ε 0 , ε ∈ {± 1 }} . Supp ose for con tradiction that x has at least κ 1 + 1 ( ε ′ )-neigh b ors in N ( ε ) ( m ) ( V ( C )). Let ( H 1 , w H 1 ) be the subgraph induced on N ( ε ) ( m ) ( V ( C )) ∩ N ( ε ′ ) ( x ). By Lemma 2.5 , the graph ( H 1 , w H 1 ) contains a ( K m , +1). By Lemma 2.3 , the v ertex x ∈ N ( ε · ε ′ ) ( m ) ( V ( C )). Note that ε · ε ′ ∈ { 0 , − ε 0 } , then ε · ε ′  = ε 0 , which con tradicts that x ∈ N ( ε 0 ) ( m ) ( V ( C )). This shows the statement (i) holds. (ii) Suppose for con tradiction that N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )) con tains at least κ 1 + 1 vertices, for some ε 1 , ε 2 ∈ {± 1 } . Let ( H 2 , w H 2 ) denote the subgraph induced on N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )). By Lemma 2.5 , the graph ( H 2 , w H 2 ) contains a ( K m , +1), denote by D . By Lemma 2.2 , each v ertex of D has at most 2 m − 2 non-( ε 1 )-neigh b ors in C 1 . Let U := { x ∈ V ( C 1 ) | V ( D ) ⊆ N ( ε 1 ) ( x ) } , then | U | ≥ n − (2 m − 2) m ≥ 4 m . Since V ( D ) ⊆ N ( ε 2 ) ( m ) ( V ( C 2 )), we find that, b y Lemma 2.3 , the vertex x ∈ N ( ε 1 · ε 2 ) ( m ) ( V ( C 2 )) for an y x ∈ U . Th us, | N ( ε 1 · ε 2 ) ( m ) ( V ( C 2 )) ∩ V ( C 1 ) | ≥ | U | ≥ 4 m . Since ε 1 , ε 2 ∈ {± 1 } and ε 1 · ε 2 ∈ {± 1 } , by Lemma 2.4 , we obtain that N (+1) ( m ) ( V ( C 1 )) = N ( ε 1 · ε 2 ) ( m ) ( V ( C 2 )) and N ( − 1) ( m ) ( V ( C 1 )) = N ( − ε 1 · ε 2 ) ( m ) ( V ( C 2 )). This contradicts that N (+1) ( m ) ( V ( C 1 )) ∪ N ( − 1) ( m ) ( V ( C 1 ))  = N (+1) ( m ) ( V ( C 2 )) ∪ N ( − 1) ( m ) ( V ( C 2 )). Hence, | N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )) | ≤ κ 1 for any ε 1 , ε 2 ∈ {± 1 } . 3 Signed graphs and quasi-p ositiv e-cliques In this section, we first introduce more definitions and notations of signed graphs, then translate the results we obtained in Subsection 2.2 to signed graphs, by considering the (0)-edges in F 3 -w eighted complete graphs as non-edges in signed graphs. 7 3.1 More definitions on signed graphs Let ( G, σ ) be a signed graph. F or { x, y } ∈ E ( G ), we say x and y are p ositively adjac ent or p ositive neighb ors if σ ( { x, y } ) = +, and w e say x and y are ne gatively adjac ent or ne gative neighb ors if σ ( { x, y } ) = − . If { x, y } ∈ E ( G ), we say x and y are non-adjac ent or non-neighb ors . F or x ∈ V ( G ), the p ositive-neighb orho o d N (+) ( x ) of x in ( G, σ ) (resp. the ne gative-neighb orho o d N ( − ) ( x ) of x in ( G, σ )) is the set of the p ositiv e- neigh b ors (resp. negativ e neigh b ors) of x . F or a positive in teger t and a v ertex subset U ⊆ V ( G ), the t -p ositive-neighb orho o d N (+) ( t ) ( x ) of U in ( G, σ ) (resp. the t -ne gative-neighb orho o d N ( − ) ( t ) ( x ) of U in ( G, σ )) is the set { x ∈ V ( G ) | x has at least t p ositiv e neighbors (resp. negativ e neigh b ors) in U } . The induced subgraph of ( G, σ ) on U ⊆ V ( G ) is denoted b y ( G, σ ) U . W e call ( G, σ ) U a p ositive clique (resp. ne gative clique ) if ( G, σ ) U is isomorphic to ( K | U | , +) (resp. ( K | U | , − )). 3.2 Quasi-p ositiv e-cliques In this subsection, w e will define quasi-positive-cliques of { e K (0) 2 m , e K ( − ) 2 m } -switc hing-free signed graphs, and sho w a few prop erties of quasi-positive-cliques. First, we state t wo results on { e K (0) 2 m , e K ( − ) 2 m } -switc hing-free signed graphs before w e define quasi-p ositiv e- cliques. Lemma 3.1. L et m ≥ 2 , n ≥ 3 m − 2 b e inte gers and ( G, σ ) b e a { e K (0) 2 m , e K ( − ) 2 m } -switching-fr e e signe d gr aph. L et C b e a p ositive clique of ( G, σ ) with at le ast n vertic es. F or any vertex x ∈ V ( G ) , one of the fol lowing holds. (i) x ∈ N (+) ( m ) ( V ( C )) , and x has at most m − 1 ne gative neighb ors and at most m − 1 non-neighb ors in C . (ii) x ∈ N ( − ) ( m ) ( V ( C )) , and x has at most m − 1 p ositive neighb ors and at most m − 1 non-neighb ors in C . (iii) x has at most m − 1 p ositive neighb ors and at most m − 1 ne gative neighb ors in C . Pr o of. It follows from Lemma 2.2 straightforw ard. Let m ≥ 2 , n ≥ 2( m 2 + m ) be integers and ( G, σ ) be a { e K (0) 2 m , e K ( − ) 2 m } -switc hing-free signed graph. Let C ( n ) denote the set { C | C is a maximal p ositiv e clique of ( G, σ ) with at least n v ertices } . The following lemma sho ws that for any pair of p ositiv e cliques in C ( n ), either there are few edges b et ween them, or they ha ve the same m -neighborho od. Lemma 3.2. L et m ≥ 2 , n ≥ 2( m 2 + m ) b e inte gers and ( G, σ ) b e a { e K (0) 2 m , e K ( − ) 2 m } -switching-fr e e signe d gr aph. F or any C, C ′ ∈ C ( n ) , one of the fol lowing holds. (i) Ther e ar e at most m − 1 vertic es of C ′ c ontaine d in N (+) ( m ) ( V ( C )) , at most m − 1 vertic es of C ′ c ontaine d in N ( − ) ( m ) ( V ( C )) , and e ach of the other vertic es of C ′ has at most 2 m − 2 neighb ors in C . (ii) N (+) ( m ) ( V ( C ′ )) ∪ N ( − ) ( m ) ( V ( C ′ )) = N (+) ( m ) ( V ( C )) ∪ N ( − ) ( m ) ( V ( C )) . Pr o of. By Lemma 2.4 , if the statemen t (i) does not hold, then either N (+) ( m ) ( V ( C ′ )) = N (+) ( m ) ( V ( C )) , N ( − ) ( m ) ( V ( C ′ )) = N ( − ) ( m ) ( V ( C )) or N (+) ( m ) ( V ( C ′ )) = N ( − ) ( m ) ( V ( C )) , N ( − ) ( m ) ( V ( C ′ )) = N (+) ( m ) ( V ( C )) . This completes the proof. 8 F or any C ∈ C ( n ), define the quasi-p ositive-clique Q ( C ) of ( G, σ ), with resp ect to the pair ( m, n ), to b e the signed graph obtained from ( G, σ ) N (+) ( m ) ( V ( C )) ∪ N ( − ) ( m ) ( V ( C )) b y switc hing on N ( − ) ( m ) ( V ( C )). Prop osition 3.3. L et m ≥ 2 , s ≥ 3 , t ≥ 2 , n ≥ max { 2( m 2 + m ) , (2 m − 1) t + 1 } b e inte gers. Ther e exists a p ositive inte ger κ 2 = κ 2 ( m, s, t ) , such that for any { e K (0) 2 m , e K ( − ) 2 m , ( K s , − ) , ( K 1 ,t , +) } -switching-fr e e signe d gr aph ( G, σ ) and any C ∈ C ( n ) , the p ositive gr aph of Q ( C ) is a κ 2 -plex, wher e Q ( C ) is the quasi-p ositive-clique with r esp e ct to C . Pr o of. Let κ 2 = κ 2 ( m, s, t ) := 4 κ 1 + 1, as defined in Lemma 2.6 . Let x b e a vertex of Q ( C ). By Lemma 2.6 (i) , there are at most 2 κ 1 non-neigh b ors of x in Q ( C ). Now w e coun t the negativ e neigh b ors of x in Q ( C ). If x ∈ N (+) ( m ) ( V ( C )), the set of negative neigh b ors of x in Q ( C ) is ( N (+) ( m ) ( V ( C )) ∩ N ( − ) ( x )) ∪ ( N ( − ) ( m ) ( V ( C )) ∩ N (+) ( x )). If x ∈ N ( − ) ( m ) ( V ( C )), the set of negative neigh b ors of x in Q ( C ) is ( N (+) ( m ) ( V ( C )) ∩ N (+) ( x )) ∪ ( N ( − ) ( m ) ( V ( C )) ∩ N ( − ) ( x )). By Lemma 2.6 (i) , there are at most 2 κ 1 negativ e neighbors of x in Q ( C ). Therefore, the p ositive graph of Q ( C ) is a κ 2 -plex. This completes the pro of. Lemma 3.2 and the follo wing lemma show that for an y pair of p ositiv e cliques in C ( n ), their quasi-p ositiv e- cliques either are the same, or hav e a b ounded in tersection. Lemma 3.4. L et m ≥ 2 , s ≥ 3 , t ≥ 2 , n ≥ max { 2( m 2 + m ) , (2 m − 1) t + 1 } b e inte gers. Ther e exists a p ositive inte ger κ 3 = κ 3 ( m, s, t ) , such that for any { e K (0) 2 m , e K ( − ) 2 m , ( K s , − ) , ( K 1 ,t , +) } -switching-fr e e signe d gr aph ( G, σ ) and any p air of C 1 , C 2 ∈ C ( n ) , if V ( Q ( C 1 ))  = V ( Q ( C 2 )) , then | N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )) | ≤ κ 3 for any ε 1 , ε 2 ∈ { + , −} . Pr o of. Let κ 3 = κ 3 ( m, s, t ) := κ 1 , as defined in Lemma 2.6 . Since V ( Q ( C 1 ))  = V ( Q ( C 2 )), we hav e N (+1) ( m ) ( V ( C 1 )) ∪ N ( − 1) ( m ) ( V ( C 1 ))  = N (+1) ( m ) ( V ( C 2 )) ∪ N ( − 1) ( m ) ( V ( C 2 )). By Lemma 2.6 (ii) , we obtain that | N ( ε 1 ) ( m ) ( V ( C 1 )) ∩ N ( ε 2 ) ( m ) ( V ( C 2 )) | ≤ κ 3 , for any ε 1 , ε 2 ∈ { + , −} . 4 Hoffman signed graphs In this section, we first introduce the definitions and k ey prop erties of Hoffman signed graphs, then construct the associated Hoffman signed graphs b y means of the induced subgraphs whic h are switc hing equiv alen t to quasi-positive-cliques. 4.1 Definitions of Hoffman signed graphs A Hoffman signe d gr aph h = ( H , σ, ℓ ) is a pair of a signed graph ( H , σ ) and a lab eling map ℓ : V ( H ) → { f , s } , satisfying that any pair of vertices with label f are non-adjacent and an y vertex with lab el f has at least one neighbor. W e call ( H , σ ) the underlying signe d gr aph of h . W e say a v ertex with label s a slim vertex , and a vertex with label f a fat vertex . Denote by V s ( h ) (resp. V f ( h )) the slim vertex set (resp. fat vertex set ) of h . If eac h slim v ertex of the Hoffman signed graph h has at least one fat neighbor, we call h fat . The slim gr aph ( H s , σ s ) of the Hoffman signed graph h is the subgraph of ( H , σ ) induced on V s ( h ). W e ma y consider a signed graph as a Hoffman signed graph with only slim vertices, and vice versa. F or the rest part of the pap er, we will not distinguish b et ween them. A Hoffman signed graph h 1 = ( H 1 , σ 1 , ℓ 1 ) is called a sub gr aph of h = ( H , σ, ℓ ), if ( H 1 , σ 1 ) is a subgraph of ( H , σ ) and ℓ 1 = ℓ | V ( H 1 ) . And we say h 1 = ( H 1 , σ 1 , ℓ 1 ) is an induc e d sub gr aph of h = ( H , σ, ℓ ), if ( H 1 , σ 1 ) is an induced subgraph of ( H , σ ) and ℓ 1 = ℓ | V ( H 1 ) . Two Hoffman signed graphs h = ( H , σ, ℓ ) and h ′ = ( H ′ , σ ′ , ℓ ′ ) are called isomorphic if there exists an isomorphism b et ween ( H , σ ) and ( H ′ , σ ′ ) that preserves the lab eling. 9 F or a Hoffman signed graph h = ( H , σ, ℓ ), let A =  A s C C T O  b e the adjacency matrix of ( H , σ ), where A s is the adjacency matrix of the slim graph of ( H , σ ). The matrix S ( h ) := A s − C C T is called the sp e cial matrix of h . The eigenvalues of a Hoffman signed graph are the eigen v alues of its special matrix. Denote by λ min ( h ) the smallest eigenv alue of the Hoffman signed graph h . Note that if a Hoffman signed graph h = ( H , σ, ℓ ) has only slim v ertices, then S ( h ) = A ( H ), and thus the sp ecial eigen v alues and the adjacency eigen v alues are also the same. W e say that Hoffman signed graph h ′ = ( H ′ , σ ′ , ℓ ′ ) can b e obtained from h = ( H , σ, ℓ ) by switching with resp ect to U for some U ⊆ V ( H ), if ( H ′ , σ ′ ) is obtained from ( H , σ ) by switching with resp ect to U and for an y x ∈ V ( H ′ ), ℓ ′ ( x ) = ℓ ( x ). W e also call h and h ′ are switching e quivalent , if h ′ can b e obtained from h by switc hing. Lemma 4.1. L et h = ( H , σ, ℓ ) b e a Hoffman signe d gr aph. If h ′ = ( H ′ , σ ′ , ℓ ′ ) is switching e quivalent to h = ( H , σ, ℓ ) , then h ′ has the same eigenvalues as h . Pr o of. Since h = ( H , σ, ℓ ) and h ′ = ( H ′ , σ ′ , ℓ ′ ) are switc hing equiv alent, w e may assume that ( H ′ , σ ′ ) is obtained from ( H , σ ) b y switc hing on U for some U ⊆ V ( H ). Define the diagonal matrix D as D xx =  − 1 , if x ∈ U 1 , otherwise . Note that  A s ( H ′ , σ ′ ) C ′ C ′ T O  = A ( H ′ , σ ′ ) = D − 1 A ( H , σ ) D = D − 1  A s ( H , σ ) C C T O  D , then A s ( H ′ , σ ′ ) = D − 1 s A s ( H , σ ) D s and C ′ C ′ T = D − 1 s C C T D s , where D s is the submatrix of D with rows and columns indexed b y vertices in V s ( h ). Thus, S ( h ′ ) = A s ( H ′ , σ ′ ) − C ′ C ′ T = D − 1 s A s ( H , σ ) D s − D − 1 s C C T D s = D − 1 s S ( h ) D s . Hence, the eigen v alues of h and h ′ are the same. Let h b e a Hoffman signed graph and n b e a p ositiv e in teger. W e denote b y G ( h , n ) the signed graph obtained b y replacing each fat vertex of h b y a positive clique ( K n , +). Hoffman in principle show ed the follo wing result in [ 10 ], and Gavrilyuk et al. [ 5 ] repro v ed it and stated it in terms of Hoffman signed graph. Theorem 4.2 ([ 5 , Theorem 4 . 2]) . F or any Hoffman signe d gr aph h and any p ositive inte ger n , λ min ( G ( h , n )) ≥ λ min ( h ) , and lim n → + ∞ λ min ( G ( h , n )) = λ min ( h ) . 4.2 Lattices Let R n b e an n -dimensional Euclidean space, equipp ed with the canonical inner pr o duct ( x , y ) := x T y . The num b er || x || 2 = ( x , x ) will be called the norm of x ∈ R n . A lattic e Λ in R n is a discrete set of vectors in R n whic h is closed under addition and subtraction. An inte gr al lattic e is a lattice in which the inner pro duct of any tw o vectors is integral. The dir e ct sum of t wo lattices Λ 1 and Λ 2 is defined if Λ 1 and Λ 2 are orthogonal, i.e., if ( x , y ) = 0 for x ∈ Λ 1 , y ∈ Λ 2 , as Λ 1 ⊕ Λ 2 = { x + y | x ∈ Λ 1 , y ∈ Λ 2 } . A lattice Λ is called irr e ducible if Λ = Λ 1 ⊕ Λ 2 implies Λ 1 = { 0 } or Λ 2 = { 0 } , and r e ducible otherwise. If X is a set of vectors in R n suc h that ( x , y ) ∈ Z for all x , y ∈ X , then Λ = { P x ∈ X α x x | α x ∈ Z } is an in tegral lattice. In this case, w e say that the lattice Λ is gener ate d b y X , and denote it b y ⟨ X ⟩ . W e sa y an in tegral lattice Λ has minimal norm t if min { ( x , x ) | x  = 0 , x ∈ Λ } = t . F or a p ositiv e integer s , an integral lattice Λ ⊂ R n is called s -inte gr able if √ s Λ can b e em b edded in the standard lattice Z k for some k ≥ n . In other words, an in tegral lattice Λ is s -in tegrable if and only if eac h v ector in Λ can be describ ed by the form 1 √ s ( x 1 , . . . , x k ) T with all x i ∈ Z in R k for some k ≥ n sim ultaneously . 10 4.3 Hoffman signed graphs and Lattices W e start with the represen tation and the reduced representation of Hoffman signed graphs. Definition 4.3. F or a Hoffman signe d gr aph h and a p ositive inte ger n , a mapping ϕ : V ( h ) → R n such that ( ϕ ( x ) , ϕ ( y )) =            m, if x = y ∈ V s ( h ) , 1 , if x = y ∈ V f ( h ) , 1 , if x, y ar e p ositively adjac ent , − 1 , if x, y ar e ne gatively adjac ent , 0 , otherwise , is c al le d a representation of h of norm m . W e denote by Λ( h , m ) the lattice generated b y { ϕ ( x ) | x ∈ V ( h ) } . Note that the isomorphism class of Λ( h , m ) dep ends only on m , and is indep enden t of ϕ , justifying the notation. F or a Hoffman signed graph h = ( H , σ, ℓ ), we define n f h ( x, y ) := X z ∈ V f ( h ) { x,z } , { y,z }∈ E ( h ) |{ z | σ ( { x, z } ) = σ ( { y , z } ) }| − X z ∈ V f ( h ) { x,z } , { y,z }∈ E ( h ) |{ z | σ ( { x, z } )  = σ ( { y , z } ) }| for x, y ∈ V s ( h ). Definition 4.4. F or a Hoffman signe d gr aph h and a p ositive inte ger n , a mapping ψ : V s ( h ) → R n such that ( ψ ( x ) , ψ ( y )) =          m − n f h ( x, y ) , if x = y , 1 − n f h ( x, y ) , if x, y ar e p ositively adjac ent , − 1 − n f h ( x, y ) , if x, y ar e ne gatively adjac ent , − n f h ( x, y ) , otherwise , is c al le d a reduced representation of h of norm m . W e denote b y Λ red ( h , m ) the lattice generated b y { ψ ( x ) | x ∈ V s ( h ) } . Note that the isomorphism class of Λ red ( h , m ) dep ends only on m , and is indep enden t of ψ , justifying the notation. Lemma 4.5. If h is a Hoffman signe d gr aph having a r epr esentation of norm m , then h has a r e duc e d r epr esentation of norm m , and Λ( h , m ) is isomorphic to Λ red ( h , m ) ⊕ Z | V f ( h ) | as a lattic e. Pr o of. Let ϕ : V ( h ) → R n b e a representation of norm m . Let U b e the subspace of R n generated by { ϕ ( x ) | x ∈ V f ( h ) } . Since ( ϕ ( x ) , ϕ ( y )) = 0 for x, y ∈ V f ( h ), w e obtain { ϕ ( x ) | x ∈ V f ( h ) } is an orthogonal basis of U . Let ψ ( x ) = ϕ ( x ) − P y ∈ V f ( h ) ( ϕ ( x ) , ϕ ( y )) ϕ ( y ) for x ∈ V s ( h ), then ψ ( x ) is a reduced representation of norm m . Note that ( ψ ( x ) , ϕ ( y )) = 0 for x ∈ V s ( h ) , y ∈ V f ( h ). Therefore, the lattice Λ( h , m ) is isomorphic to Λ red ( h , m ) ⊕ Z | V f ( h ) | . F or the unsigned case, Lemma 4.5 was shown by Jang et al. [ 11 ]. Our proof of the ab o ve lemma is essen tially iden tical to theirs. Theorem 4.6. F or a Hoffman signe d gr aph h = ( H , σ, ℓ ) with smal lest eigenvalue λ min ( h ) , the fol lowing ar e e quivalent. (i) h has a r epr esentation of norm m . (ii) h has a r e duc e d r epr esentation of norm m . (iii) λ min ( h ) ≥ − m . 11 Pr o of. It follows from Lemma 4.5 that (i) implies (ii). Let S p ( h ) be the special matrix of h . Let ψ b e a reduced representation of h of norm m . The matrix S p ( h ) + m I is the Gram matrix of the image of ψ . Hence the matrix S p ( h ) + m I is p ositive semidefinite, in other words, the matrix S p ( h ) has smallest eigen v alue at least − m . This sho ws that (ii) implies (iii). No w w e prov e that (iii) implies (i) . Assume that λ min ( h ) ≥ − m , then the matrix S p ( h ) + m I is p ositiv e semidefinite, hence has a Cholesky factorization LL T with square L . The following map φ ( x ) = ( e x , if x is fat, P y ∈ V s ( h ) L xy e y + P y ∈ V f ( h ) ,σ ( { x,y } )=+ e y − P y ∈ V f ( h ) ,σ ( { x,y } )= − e y , otherwise, is a representation of h of norm m , where { e x | x ∈ V ( h ) } is a set of standard unit v ectors. This completes the pro of. As a direct consequence of Theorem 4.6 , w e find the relationship b et ween the smallest eigenv alue of a Hoffman signed graph and the smallest eigen v alue of its induced Hoffman signed subgraph. F or the unsigned case, this result w as shown b y W o o and Neumaier [ 16 ]. Lemma 4.7. L et h b e a Hoffman signe d gr aph. If g is an induc e d Hoffman signe d sub gr aph of h , then λ min ( g ) ≥ λ min ( h ) . In p articular, λ min ( H s , σ s ) ≥ λ min ( h ) , wher e ( H s , σ s ) is the slim gr aph of h . Pr o of. Assume that λ min ( h ) = − m . By Theorem 4.6 , h has a represen tation of norm m . Since g is an induced Hoffman signed subgraph of h , then g also has a represen tation of norm m . Thus, λ min ( g ) ≥ − m = λ min ( h ), b y Theorem 4.6 . As the slim graph ( H s , σ s ) is an induced Hoffman signed subgraph of h , w e obtain that λ min ( H s , σ s ) ≥ λ min ( h ). Lemma 4.8. L et h b e a Hoffman signe d gr aph with smal lest eigenvalue λ min ( h ) , and let m b e an inte ger with m ≥ − λ min ( h ) . The inte gr al lattic e Λ red ( h , m ) is s -inte gr able if and only if the inte gr al lattic e Λ( h , m ) is s -inte gr able. Pr o of. By Lemma 4.5 and the definition of integrabilit y of lattices, we obtain Λ red ( h , m ) is s -in tegrable if and only if the in tegral lattice Λ( h , m ) is s -in tegrable. Corollary 4.9. L et h b e a Hoffman signe d gr aph with smal lest eigenvalue λ min ( h ) , and let ( H s , σ s ) b e the slim gr aph of h with smal lest eigenvalue λ min ( H s , σ s ) . Supp ose ⌊ λ min ( h ) ⌋ = ⌊ λ min ( H s , σ s ) ⌋ holds. The gr aph ( H s , σ s ) is s -inte gr able if the inte gr al lattic e Λ( h , −⌊ λ min ( h ) ⌋ ) is s -inte gr able. Pr o of. This follo ws from Lemma 4.8 immediately . 4.4 Asso ciated Hoffman signed graphs Let m ≥ 2 , n ≥ 2( m 2 + m ) be integers and ( G, σ ) be a { e K (0) 2 m , e K ( − ) 2 m } -switc hing-free signed graph. Let C ( n ) denote the set of maximal positive cliques of ( G, σ ) with at least n v ertices. F or C ∈ C ( n ), recall that N (+) ( m ) ( V ( C )) (resp. N ( − ) ( m ) ( V ( C ))) denotes the m -p ositiv e-neigh b orhoo d (resp. the m -negativ e-neighborho od) of C . F or con venience, let N ( m ) ( V ( C )) := N (+) ( m ) ( V ( C )) ∪ N ( − ) ( m ) ( V ( C )). In Subsection 3.2 , w e ha v e seen that if switching the subgraph induced on N ( m ) ( V ( C )) with respect to N ( − ) ( m ) ( V ( C )), we obtain the quasi-positive- clique with resp ect to C . Now w e are ready to define asso ciated Hoffman signed graphs. Definition 4.10. L et m ≥ 2 , n ≥ 2( m 2 + m ) b e inte gers and ( G, σ ) b e a { e K (0) 2 m , e K ( − ) 2 m } -switching-fr e e signe d gr aph. L et C ( n ) denote the set of maximal p ositive cliques of ( G, σ ) with at le ast n vertic es. L et { N ( m ) ( V ( C )) | C ∈ C ( n ) } = { N ( m ) ( V ( C 1 )) , . . . , N ( m ) ( V ( C r )) } . The Hoffman signe d gr aph satisfying the fol lowing c onditions is an asso ciated Hoffman signed graph of ( G, σ ) , with r esp e ct to the p air ( m, n ) , denote d by g = g ( G, σ, m, n ) . (i) V s ( g ) = V ( G ) , and the slim gr aph of g is ( G, σ ) . 12 (ii) V f ( g ) = { F 1 , F 2 , . . . , F r } , and e ach fat vertex F i is p ositively adjac ent to e ach vertex in N (+) ( m ) ( V ( C i )) , ne gatively adjac ent to e ach vertex in N ( − ) ( m ) ( V ( C i )) , and non-adjac ent to the other vertic es of V ( G ) , for i = 1 , . . . , r . Remark 4.11 is important to help us understand asso ciated Hoffman signed graphs better. Remark 4.11. (i) In the ab ove definition, the set { C 1 , . . . , C r } may mer ely b e a pr op er subset of C ( n ) , as ther e may b e two distinct p ositive cliques C, C ′ ∈ C ( n ) , such that N ( m ) ( V ( C )) = N ( m ) ( V ( C ′ )) , by L emma 3.2 . (ii) It is p ossible that ther e exists another subset { C ′ 1 , . . . , C ′ r } of C ( n ) , such that N ( m ) ( V ( C 1 )) = N ( m ) ( V ( C ′ 1 )) for i = 1 , . . . , r , then we obtain another asso ciate d Hoffman signe d gr aph g ′ of ( G, σ ) . Note that g ′ may b e differ ent fr om g . However, they have the same sp e cial matrix. Next prop osition connects signed graphs and Hoffman signed graphs, which is crucial in next section. Prop osition 4.12. L et m ≥ 2 , v f ≥ 0 , v s ≥ 0 , p ≥ 0 b e inte gers. Ther e exists a p ositive inte ger q = q ( m, v f , v s , p ) ≥ 2( m 2 + m ) , such that, for any signe d gr aph ( G, σ ) , any inte ger n ≥ q and any Hoffman signe d gr aph h with at most v f fat vertic es and at most v s slim vertic es, the signe d gr aph G ( h , p ) is isomorphic to an induc e d sub gr aph of ( G, σ ) , pr ovide d that the fol lowing hold. (i) The signe d gr aph ( G, σ ) is { e K (0) 2 m , e K ( − ) 2 m } -switching-fr e e. (ii) The Hoffman signe d gr aph h is isomorphic to an induc e d sub gr aph of an asso ciate d Hoffman signe d gr aph g ( G, σ, m, n ) . Pr o of. Let q = max 0 ≤ v ′ f ≤ v f , 0 ≤ v ′ s ≤ v s { q ′ ( m, v ′ f , v ′ s , p ) } , where q ′ ( m, 0 , v ′ s , p ) := 2( m 2 + m ), and q ′ ( m, v ′ f , v ′ s , p ) := max { q ′ ( m, v ′ f − 1 , v ′ s , p + 2 m − 2) , (2 m − 1) v ′ s + (2 m − 2)( v ′ f − 1) p + p } for v ′ f ≥ 1. Let n ≥ q , ( G, σ ) b e a signed graph satisfying the giv en conditions. Let g := g ( G, σ, m, n ) b e an asso ciated Hoffman signed graph of ( G, σ ). Let h b e an induced Hoffman subgraph of g with v ′ f fat vertices and v ′ s slim vertices where v ′ f ≤ v f and v ′ s ≤ v s . W e will pro ve this prop osition b y induction on v ′ f . When v ′ f = 0, h con tains no fat v ertices. Thus, G ( h , p ) is the slim graph of h . It is clear that G ( h , p ) is an induced subgraph of ( G, σ ). No w we consider the case v ′ f ≥ 1. Let V f ( h ) = { F 1 , F 2 , . . . , F v ′ f } b e the fat v ertex set of h and V s ( h ) b e the slim vertex set of h . By the definition of asso ciated Hoffman signed graphs, there exists C i ∈ C ( n ), such that the set N ( m ) ( V ( C i )) is the neigh b orhoo d of F i in g , for i = 1 , 2 , . . . , v ′ f . Let h 1 denote the subgraph of h obtained b y deleting the fat vertex F v ′ f . By the induction hypothesis, the signed graph G ( h 1 , p + 2 m − 2) is isomorphic to an induced subgraph of ( G, σ ). Denote by D i the p ositiv e clique in G ( h 1 , p + 2 m − 2) replacing the fat vertex F i for i = 1 , 2 , . . . , v ′ f − 1. By Lemma 3.1 , for x ∈ V s ( h ), if x is p ositiv ely adjacent to F v ′ f , then x has at most 2 m − 2 non-p ositiv e neigh b ors in C v ′ f ; if x is negatively adjacen t to F v ′ f , then x has at most 2 m − 2 non-negative neigh b ors in C v ′ f ; if x not adjacent to F , then it has at most 2 m − 2 neighbors in C v ′ f . Therefore, we are able to obtain a p ositiv e clique D v ′ f con taining at least n − v ′ s − (2 m − 2) v ′ s v ertices, suc h that V s ( h ) ∩ V ( D v ′ f ) = ∅ , and for x ∈ V s ( h ), if x is p ositiv ely adjacent to F v ′ f , then x is p ositiv ely adjacent to each v ertex of D v ′ f ; if x is negativ ely adjacen t to F v ′ f , then x is negatively adjacen t to each v ertex of D v ′ f ; if x is non-adjacent to F v ′ f , then x is non-adjacen t to each v ertex of D v ′ f . By Lemma 3.2 , | V ( D i ) − N ( m ) ( V ( C v ′ f )) | ≥ p + 2 m − 2 − 2( m − 1) = p , th us there exists a p ositiv e clique of order p with vertex set in V ( D i ) − N ( m ) ( V ( C v ′ f )), denoted by D ′ i . Also, | V ( D v ′ f ) − { x ∈ V ( D v ′ f ) | x has at least one neigh b or in S v ′ f − 1 i =1 V ( D ′ i ) }| ≥ | V ( D v ′ f ) | − P v ′ f − 1 i =1 (2 m − 2) | V ( D ′ i ) | ≥ n − v ′ s − (2 m − 2) v ′ s − ( v ′ f − 1)(2 m − 2) p ≥ p . Hence, there exist a p ositiv e clique of order p in D v ′ f , denoted by D ′ v ′ f , such that eac h v ertex in D ′ v ′ f is not adjacent to any v ertex in S v ′ f − 1 i =1 V ( D ′ i ). 13 Note that D ′ 1 , D ′ 2 , . . . , D ′ v ′ f − 1 are isolated positive cliques in ( G, σ ) with p v ertices. Thus, the subgraph of ( G, σ ) induced on V s ( h ) ∪ ( S v ′ f i =1 V ( D ′ i )) is isomorphic to G ( h , p ). 5 Pro ofs of Theorem 1.1 (ii) and Theorem 1.3 In this section, w e will prov e Theorem 1.1 (ii) and Theorem 1.3 . First we sho w the following theorem. Theorem 5.1. L et m ≥ 2 , s ≥ 3 , t ≥ 2 b e inte gers. Ther e exist two p ositive inte gers q = q ( m, t ) , κ 4 = κ 4 ( m, s, t ) such that for any { e K (0) 2 m , e K ( − ) 2 m , ( K s , − ) , ( K 1 ,t , +) } -switching-fr e e signe d gr aph ( G, σ ) and any inte ger n ≥ q , if g = g ( G, σ, m, n ) is an asso ciate d Hoffman signe d gr aph of ( G, σ ) with r esp e ct to the p air ( m, n ) , then the sub gr aphs N 1 , . . . , N r ( r = | V f ( g ) | ) induc e d on the neighb ors of e ach fat vertex of g satisfy the fol lowing. (i) Each vertex in ( G, σ ) is c ontaine d in at most t − 1 N i ’s. (ii) The induc e d sub gr aph N i is switching e quivalent to a signe d gr aph whose p ositive gr aph is a κ 4 -plex, for i = 1 , . . . , r . (iii) The interse ction V ( N i ) ∩ V ( N j ) c ontains at most κ 4 vertic es, for 1 ≤ i < j ≤ r . (iv) The sub gr aph ( G ′ , σ ′ ) has maximum valency at most R ( n, s, t ) − 1 , wher e G ′ = ( V ( G ) , E ( G ) \ S r i =1 E ( N i )) and σ ′ = σ | E ( G ′ ) . Pr o of. Let q := max { 2( m 2 + m ) , 4( t − 1)( m − 1) + m } and κ 4 := max { κ 2 , 4 κ 3 } , where κ 2 and κ 3 are as defined in Prop osition 3.3 and Lemma 3.4 resp ectiv ely . F or any n ≥ q , denote by C ( n ) the set { C | C is a maximal p ositiv e clique of ( G, σ ) with at least n v ertices } , and for C ∈ C ( n ), let N ( m ) ( V ( C )) = N (+) ( m ) ( V ( C )) ∪ N ( − ) ( m ) ( V ( C )). Let { N ( m ) ( V ( C 1 )) , . . . , N ( m ) ( V ( C r )) } = { N ( m ) ( V ( C )) | C ∈ C ( n ) } . Let N i := ( G, σ ) N ( m ) ( V ( C i )) denote the subgraph of ( G, σ ) induced on N ( m ) ( V ( C i )) for i = 1 , . . . , r . (i) Supp ose that there exists a v ertex x suc h that x is con tained in at least t N i ’s. Without loss of generalit y , w e may assume that x ∈ T t i =1 N ( m ) ( V ( C i )). Let U i = V ( C i ) ∩ { y | y is adjacent to x } − S j  = i 1 ≤ j ≤ t N ( m ) ( V ( C j )). By Lemma 3.1 and Lemma 3.2 , | U i | ≥ n − ( m − 1) − ( t − 1)(2 m − 2) ≥ 4( t − 1)( m − 1) + m − ( m − 1) − ( t − 1)(2 m − 2) ≥ ( t − 1)(2 m − 2) + 1, for 1 ≤ i ≤ t . T ake a vertex y 1 from U 1 . F or i = 2 , . . . , t , there exists y i ∈ U i − S i − 1 j =1 { y | y is adjacent to y j } , since | U i − S i − 1 j =1 { y | y is adjacent to y j }| ≥ ( t − 1)(2 m − 2) + 1 − ( i − 1)(2 m − 2) ≥ 1. The subgraph induced on { x, y 1 , y 2 , . . . , y t } is switching equiv alen t to ( K 1 ,t , +), which is a contradiction. (ii) Let Q i = Q ( C i ) b e the quasi-positive-clique obtained from N i b y switc hing on N ( − ) ( m ) ( V ( C i )). By Prop osition 3.3 , the p ositiv e graph of Q i is a κ 4 -plex, as κ 4 ≥ κ 2 . Therefore, (ii) holds. (iii) By Lemma 3.4 , w e ha ve | V ( N i ) ∩ V ( N j ) | ≤ P ε 1 ,ε 2 ∈{ + , −} | V ( N ( ε 1 ) ( m ) ( V ( C i )) ∩ V ( N ( ε 2 ) ( m ) ( V ( C j )) | ≤ 4 κ 3 ≤ κ 4 for 1 ≤ i < j ≤ r . (iv) Supp ose that there exists a vertex z in ( G ′ , σ ′ ) with v alency at least R ( n, s, t ). Let ( H , σ H ) denote the subgraph of ( G, σ ) induced on the neigh b ors of z in ( G ′ , σ ′ ). Since ( G, σ ) is { ( K s , − ), ( K 1 ,t , +) } -switc hing-free, the subgraph ( H , σ H ) con tains a ( K n , +), b y Theorem 2.1 . Notice that ( K n , +) is contained in some maximal p ositiv e clique C ∈ C ( n ) in ( G, σ ). Since { N ( m ) ( V ( C 1 )) , . . . , N ( m ) ( V ( C r )) } = { N ( m ) ( V ( C )) | C ∈ C ( n ) } , there exists i suc h that N ( m ) ( C ) = N ( m ) ( C i ). Note that z is adjacent to eac h vertex in this ( K n , +). Thus, b y Lemma 3.1 , z ∈ N ( m ) ( C ) = N ( m ) ( C i ), and the edges b et ween z and this ( K n , +) are in N i , not in ( G ′ , σ ′ ). This gives a con tradiction and (iv) holds. No w w e prov e Theorem 1.1 (ii). Pro of of Theorem 1.1 (ii) . When t = 1, ( G, σ ) is an empt y graph, as it is ( K 2 , − )-switc hing-free. In this case, λ min ( G, σ ) = 0. Now w e may assume t ≥ 2. Let κ := max { κ 4 ( t, t + 1 , t ) , R ( q ( t, t ) , t + 1 , t ) − 1 } , where q ( t, t ) , κ 4 ( t, t + 1 , t ) are as defined in Theorem 5.1 . Let g := g ( G, σ, t, q ( t, t )) b e an asso ciated Hoffman graph 14 of ( G, σ ) with resp ect to the pair ( t, q ( t, t )) and N 1 , N 2 , . . . , N r b e the subgraphs induced on the neigh b ors of eac h fat v ertex of g , where r = | V f ( g ) | . Let S := S ( g ) b e the special matrix of g . By Lemma 4.7 , w e hav e λ min ( G, σ ) ≥ λ min ( g ). Giv en a v ertex x of ( G, σ ), assume that N 1 , . . . , N p are all subgraphs among N 1 , . . . , N r whic h con tain x . By Theorem 5.1 , p ≤ t − 1. Now w e are going to lo ok at the v ertices y of ( G, σ ) suc h that S xy  = 0. Let U = { y | y ∈ V s ( g ) − { x } , S xy  = 0 } , U 0 = { y ∈ U | y and x ha v e no common fat neighbors } , U 1 = { y ∈ U | y and x ha v e exactly one common fat neighbor } , U ≥ 2 = { y ∈ U | y and x ha v e at least t wo common fat neighbors } . F or any y ∈ U 0 , | S xy | = 1 holds, and b y Theorem 5.1 (iv) , we ha ve | U 0 | ≤ R ( q ( t, t ) , t + 1 , t ) − 1 ≤ κ . F or an y y ∈ U 1 , | S xy | ≤ 2 holds. Note that | U 1 | = | U 1 ∩ ( S p i =1 V ( N i )) | ≤ P p i =1 | U i ∩ V ( N i ) | . Let Q i denote the corresp onding quasi-positive-clique with resp ect to N i . F or an y y ∈ U 1 ∩ V ( N i ), S xy  = 0 implies that x and y are not p ositiv ely adjacent in Q i . By Theorem 5.1 (ii) , w e ha ve | U 1 ∩ V ( N 1 ) | ≤ κ 4 − 1 ≤ κ − 1. Hence, | U 1 | ≤ p ( κ − 1) ≤ ( t − 1)( κ − 1). F or any y ∈ U ≥ 2 , since x has at most t − 1 fat neigh b ors, | S xy | ≤ 1 + ( t − 1) = t holds. Moreov er, if y ∈ U ≥ 2 , then x, y ∈ V ( N i ) ∩ V ( N j ) for some 1 ≤ i < j ≤ p . By Theorem 5.1 (iii) , | V ( N i ) ∩ V ( N j ) | ≤ κ 4 ≤ κ . Thus, | U ≥ 2 | ≤  p − 1 2  κ 4 ≤  t − 1 2  κ . Therefore, X y | S xy | = | S xx | + X y ∈ U 0 | S xy | + X y ∈ U 1 | S xy | + X y ∈ U ≥ 2 | S xy | ≤ t − 1 + | U 0 | + 2 | U 1 | + t · | U ≥ 2 | ≤ t − 1 + κ + 2( t − 1)( κ − 1) + t  t − 1 2  κ = t ( t − 1)( t − 2) κ 2 + 2 tκ − κ − t + 1 , and λ min ( G, σ ) ≥ λ min ( g ) ≥ − max x      X y S xy      ≥ − max x X y | S xy | ≥ − t ( t − 1)( t − 2) κ 2 − 2 tκ + κ + t − 1 . This completes the proof. F or signed graph with fixed smallest eigenv alue, w e obtain the follo wing result. Theorem 5.2. L et λ < − 1 b e a r e al numb er. Ther e exist p ositive inte gers m λ , q λ such that for any inte ger n ≥ q λ , a p ositive inte ger d = d ( λ, n ) satisfying the fol lowing exists. F or any signe d gr aph ( G, σ ) with smal lest eigenvalue at le ast λ and minimum valency at le ast d ( λ, n ) , if g ( G, σ, m λ , n ) is an asso ciate d Hoffman signe d gr aph of ( G, σ ) with r esp e ct to the p air ( m λ , n ) , then the sub gr aphs N 1 , N 2 , . . . , N r ( r = | V f ( g ) | ) induc e d on the neighb ors of e ach fat vertex of g have the fol lowing pr op erties. (i) Each vertex in ( G, σ ) is c ontaine d in at le ast one and at most ⌊− λ ⌋ N i ’s. (ii) The induc e d sub gr aph N i is switching e quivalent to a signe d gr aph whose p ositive gr aph is a ⌊ λ 2 + 2 λ + 2 ⌋ - plex, for i = 1 , 2 , . . . , r . (iii) The interse ction V ( N i ) ∩ V ( N j ) c ontains at most 4 ⌊− λ ⌋ − 4 vertic es for 1 ≤ i < j ≤ r . (iv) The sub gr aph ( G ′ , σ ′ ) has maximum valency at most d ( λ, n ) − 1 , wher e G ′ = ( V ( G ) , E ( G ) \ S r i =1 E ( N i )) and σ ′ = σ | E ( G ′ ) . 15 Pr o of. Considering lim m → + ∞ λ min ( e K (0) 2 m ) = lim m → + ∞ λ min ( e K ( − ) 2 m ) = −∞ , let m λ = min { m | λ min ( e K (0) 2 m ) < λ and λ min ( e K ( − ) 2 m ) < λ } . T o obtain q λ , we need to introduce a few families of Hoffman signed graphs with smallest eigen v alue less than λ . Let h 0 b e the fat Hoffman signed graph with | V s ( h 0 ) | = 1 and | V f ( h 0 ) | = ⌊− λ ⌋ + 1, such that the unique slim vertex is p ositiv ely adjacent to all the fat vertices of h 0 . Let H 1 b e the set of fat Hoffman signed graphs with exactly ⌊ ( λ + 1) 2 ⌋ + 1 slim v ertices and one fat vertex, suc h that the unique fat vertices is p ositiv ely adjacent to all slim v ertices and there exists at least one slim v ertex whic h is non-positively adjacent to all of the other slim vertices. Let H 2 b e the set of fat Hoffman signed graphs with exactly ⌊− λ ⌋ slim v ertices and t wo fat v ertices, such that eac h fat vertex is p ositiv ely adjacen t to all of the slim vertices. Let Σ 0 = { h | h is switc hing equiv alent to h 0 } , Σ i = { h | h is switc hing equiv alent to some member in H i } for i = 1 , 2. Now w e sho w that for any h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 , λ min ( h ) < λ holds. F or an y h ∈ Σ 0 , w e hav e λ min ( h ) = λ min ( h 0 ) ≤ − ( ⌊− λ ⌋ + 1) < λ , as h and h 0 are switc hing equiv alen t, by Lemma 4.1 . F or any h ∈ Σ 1 , assume that h is switching equiv alen t to h 1 ∈ H 1 , where x is a slim vertex of h 1 whic h is non-p ositiv ely adjacen t to all of the other slim v ertices. It follo ws that S ( h 1 ) xy =  − 1 , if x and y are non-adjacent, − 2 , otherwise, for y ∈ V s ( h 1 ). Note that S ( h 1 ) y y = − 1 and S ( h 1 ) y y ′ ≤ 0 for any y , y ′ ∈ V s ( h 1 ), as h 1 has a unique fat v ertex p ositiv ely adjacen t to all the slim v ertices of h 1 . By Perron-F rob enius Theorem, we hav e λ max ( − S ( h 1 )) ≥ λ max ( A ( K 1 , ⌊ ( λ +1) 2 ⌋ +1 , +) + I ) = p ⌊ ( λ + 1) 2 ⌋ + 1 + 1 > − λ . Th us, by Lemma 4.1 , λ min ( h ) = λ min ( h 1 ) = − λ max ( − S ( h 1 )) < λ . F or an y h ∈ Σ 2 , assume that h is switching equiv alen t to h 2 ∈ H 2 . Since h 2 has exactly t w o fat vertices and each slim vertex is p ositiv ely adjacent to b oth of the fat vertices, the sp ecial matrix S ( h 2 ) satisfies that S ( h 2 ) xy =        − 2 , if x = y , − 1 , if x and y are p ositiv ely adjacent, − 2 , if x and y are non-adjacent, − 3 , if x and y are negatively adjacen t. Applying the Perron-F rob enius Theorem again yields λ max ( − S ( h 2 )) ≥ λ max ( A ( K ⌊− λ ⌋ , +) + 2 I ) ≥ ⌊− λ ⌋ + 1 > − λ , then b y Lemma 4.1 again, λ min ( h ) = λ min ( h 2 ) = − λ max ( − S ( h 2 )) < λ . By Theorem 4.2 , for any h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 , there exists a p ositiv e integer p h suc h that λ min ( G ( h , p h )) < λ . Let v f = max {| V f ( h ) | | h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 } , v s = max {| V s ( h ) | | h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 } and p = max { p h | h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 } . Let q = q ( m λ , v f , v s , p ) b e the in teger as defined in Prop osition 4.12 . Let q λ = max { 2( m 2 λ + m λ ) , (2 m λ − 1) t + 1 , q } . F or an y integer n ≥ q λ , let d ( λ, n ) = R ( n, 2 − ⌊ λ ⌋ , ⌊ λ − 1 ⌋ 2 ). Let g := g ( G, σ, m λ , n ) be an asso ciated Hoffman graph of ( G, σ ) with resp ect to the pair ( m λ , n ) and N 1 , N 2 , . . . , N r b e the subgraphs induced on the neigh b ors of eac h fat v ertex of g , where r = | V f ( g ) | . Since λ min ( G, σ ) ≥ λ , ( G, σ ) is { e K (0) 2 m λ , e K ( − ) 2 m λ } -switc hing- free and max { λ min ( e K (0) 2 m λ ) , λ min ( e K ( − ) 2 m λ ) } < λ and λ min ( G ( h , p )) ≤ λ min ( G ( h , p h )) < λ , then ( G, σ ) do es not con tain G ( h , p ) as an induced subgraph for an y h ∈ Σ 0 ∪ Σ 1 ∪ Σ 2 . Thus, g do es not contain an y Hoffman signed graph in Σ 0 ∪ Σ 1 ∪ Σ 2 as an induced subgraph, b y Prop osition 4.12 . (i) F or an arbitrary v ertex z 0 of ( G, σ ), let ( H 1 , σ H 1 ) denote the subgraph of ( G, σ ) induced on the neigh b ors of z 0 in ( G, σ ). Note that ( G, σ ) is { ( K 2 −⌊ λ ⌋ , − ) , ( K 1 , ⌊ λ − 1 ⌋ 2 , +) } -switc hing-free, as the smallest eigen v alues of ( K 2 −⌊ λ ⌋ , − ) and ( K 1 , ⌊ λ − 1 ⌋ 2 , +) are less than λ . Since the v alency of z 0 is at least R ( n, 2 − ⌊ λ ⌋ , ⌊ λ − 1 ⌋ 2 ), b y Theorem 2.1 , the subgraph ( H 1 , σ H 1 ) contains a ( K n , +). Notice that this ( K n , +) is contained in some maximal p ositiv e clique C ∈ C ( n ) in ( G, σ ). Since { N ( m ) ( V ( C 1 )) , . . . , N ( m ) ( V ( C r )) } = { N ( m ) ( V ( C )) | C ∈ C ( n ) } , there exists i suc h that z 0 ∈ N ( m ) ( V ( C )) = V ( N i ). Supp ose that z 0 is contained in at least ⌊− λ ⌋ + 1 N i ’s, then z 0 has at least ⌊− λ ⌋ + 1 fat neighbors in g . Thus, the subgraph induced on z 0 and its ⌊− λ ⌋ + 1 fat neigh b ors is contained in Σ 0 , which giv es a con tradiction. This shows that each v ertex of ( G, σ ) is contained in at most ⌊− λ ⌋ N i ’s. (ii) Let Q i = Q ( C i ) b e the quasi-positive-clique obtained from N i b y switching on N ( − ) ( m ) ( V ( C i )) and F i b e the corresponding fat v ertex of N i . Supp ose that there exists a v ertex z 1 in Q i suc h that z 1 has at least 16 ⌊ ( λ + 1) 2 ⌋ + 1 non-p ositiv e neighbors in Q i . Let U b e a set of non-p ositiv e neigh b ors of z 1 in Q i suc h that | U | = ⌊ ( λ + 1) 2 ⌋ + 1. Th us, the subgraph of g induced on { z 1 , F i } ∪ U is contained in Σ 1 . This gives a con tradiction and then (ii) holds. (iii) Supp ose that there exists a pair i  = j ∈ { 1 , . . . , r } suc h that | V ( N i ) ∩ V ( N j ) | ≥ 4 ⌊− λ ⌋ − 3, then there exists a pair of ε i , ε j ∈ { + , −} suc h that | N ( ε i ) ( m ) ( V ( C i )) ∩ N ( ε j ) ( m ) ( V ( C j )) | ≥ ⌈ 4 ⌊− λ ⌋− 3 4 ⌉ = ⌊− λ ⌋ . Note that the subgraph of g induced on ( N ( ε i ) ( m ) ( V ( C i )) ∩ N ( ε j ) ( m ) ( V ( C j )) ∪ { F i , F j } is contained in Σ 2 , which giv es a con tradiction. Thus, there are at most 4 ⌊− λ ⌋− common v ertices in both N i and N j , for any 1 ≤ i < j ≤ r . (iv) Supp ose that there exists a vertex z 2 in ( G ′ , σ ′ ) with v alency at least d ( λ, n ) ≥ R ( n, 2 − ⌊ λ ⌋ , ⌊ λ − 1 ⌋ 2 ). Let ( H 2 , σ H 2 ) denote the subgraph of ( G, σ ) induced on the neighbors of z 2 in ( G ′ , σ ′ ). Since ( G, σ ) is { ( K 2 −⌊ λ ⌋ , − ) , ( K 1 , ⌊ λ − 1 ⌋ 2 , +) } -switc hing-free, the subgraph ( H 2 , σ H 2 ) contains a ( K n , +), by Theorem 2.1 . Notice that this ( K n , +) is contained in some maximal p ositiv e clique C ′ ∈ C ( n ) in ( G, σ ). Since { N ( m ) ( V ( C 1 )) , . . . , N ( m ) ( V ( C r )) } = { N ( m ) ( V ( C )) | C ∈ C ( n ) } , there exists j suc h that N ( m ) ( C ′ ) = N ( m ) ( C j ). Note that z 2 is adjacent to each vertex in this ( K n , +). Thus, by Lemma 3.1 , z 2 ∈ N ( m ) ( C ′ ) = N ( m ) ( C j ), and the edges b et ween z 2 and this ( K n , +) are in N j , not in ( G ′ , σ ′ ). This gives a contradiction and (iv) holds. This completes the proof. No w w e prov e Theorem 1.3 . Pro of of Theorem 1.3 . Note that for λ < − 1, Theorem 1.3 is a consequence of Theorem 5.2 . No w w e may assume that λ = − 1. Note that λ min ( K 3 , − ) = − 2 and λ min ( K 1 , 2 , +) = − √ 2 , then ( G, σ ) is { ( K 3 , − ) , ( K 1 , 2 , +) } -switc hing-free. This means that ( G, σ ) is switching equiv alen t to a signed graph where eac h comp onen t is a p ositiv e clique. Let d λ = 1 and N 1 , . . . , N r b e all the distinct components of ( G, σ ), where r is a positive in teger. Note that N i is switching equiv alen t to a p ositiv e clique ( K | N i | , +) for i = 1 , . . . , r . Hence, the four statemen ts hold. This completes the proof. 6 Signed graphs with smallest eigen v alue greater than − 1 − √ 2 In this section, we focus on signed graphs with smallest eigenv alue greater than − 1 − √ 2 and give a proof of Theorem 1.6 . Let λ ≤ − 1 be a real num b er, a minimal forbidden fat Hoffman signe d gr aph f for λ is a fat Hoffman signed graph with smallest eigenv alue less than λ suc h that an y proper fat induced Hoffman signed subgraph of f has smallest eigen v alue at least λ . Denote b y F ( λ ) the set of pairwise non-isomorphic minimal forbidden fat Hoffman signed graphs for λ . W e will start with a result on minimal forbidden fat Hoffman signed graphs, whic h will be used later in the pro of of Theorem 1.6 . T o prov e it, we need the following lemma, whic h was shown b y Gavrilyuk et al. [ 5 ]. Lemma 6.1 ([ 5 , Prop osition 4 . 7]) . Each c onne cte d signe d gr aph with smal lest eigenvalue gr e ater than − √ 2 is switching e quivalent to a p ositive clique. The following theorem sho ws that the set F ( − 2) is finite. Theorem 6.2. L et f ∈ F ( − 2) and S ( f ) b e the sp e cial matrix of f . (i) If f c ontains a slim vertex with at le ast 2 fat neighb ors, then S ( f ) is ( − 3) or  − 2 a 2 a 2 a 1  , wher e a 1 , a 2 ar e inte gers such that a 1 ∈ {− 1 , − 2 } and 1 ≤ | a 2 | ≤ 1 − a 1 . (ii) If e ach slim vertex of f has exactly one fat neighb or, then S ( f ) is one of the fol lowing.  − 1 2 2 − 1  ,  − 1 − 2 − 2 − 1  ,   − 1 − 1 1 − 1 − 1 1 1 1 − 1   ,   − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1   and 17   − 1 1 1 1 − 1 0 1 0 − 1   ,   − 1 − 1 1 − 1 − 1 0 1 0 − 1   ,   − 1 − 1 − 1 − 1 − 1 0 − 1 0 − 1   . In p articular, λ min ( f ) ≤ − 1 − √ 2 and the set F ( − 2) is finite. Pr o of. (i) Assume that f con tains a slim v ertex with at least 2 fat neighbors. Since f ∈ F ( − 2), we ha ve the sp ecial matrix S ( f ) is ( − 3) or  − 2 a 2 a 2 a 1  , where a 1 , a 2 are integers suc h that a 1 ∈ {− 1 , − 2 } and 1 ≤ | a 2 | ≤ 1 − a 1 . By direct computation, w e obtain that λ min ( f ) < − 1 − √ 2. (ii) Note that all of the diagonal entries of S ( f ) are − 1, as each slim vertex of f has exactly one fat neigh b or. If S ( f ) contains an en try with absolute v alue at least 2, then S ( f ) =  − 1 b b − 1  , where b ∈ {− 2 , 2 } and λ min ( f ) = − 3. If all the en tries of S ( f ) ha ve absolute v alue at most 1, then there exists a signed graph ( H , τ ) with adjacency matrix A ( H , τ ) = S ( f ) + I . Since the smallest eigenv alue of ( H , τ ) satisfies that λ min ( H , τ ) = λ min ( f ) + 1 < − 1, then we obtain that λ min ( H , τ ) ≤ − √ 2 , by Lemma 6.1 . Note that for any signed graph with smallest eigenv alue at most − √ 2 , it contains an induced subgraph switc hing equiv alen t to either ( K 3 , − ) or ( K 1 , 2 , +). Thus, ( H , τ ) is switching equiv alen t to either ( K 3 , − ) or ( K 1 , 2 , +) and the sp ecial matrix S ( f ) = A ( H , τ ) − I . By direct computation, λ min ( f ) ≤ λ min ( H , τ ) − 1 ≤ − 1 − √ 2. Since for any f ∈ F ( − 2), f has at most 3 slim vertices and at most 4 fat vertices, which implies that F ( − 2) is finite. This completes the pro of. Corollary 6.3. L et h b e a fat Hoffman signe d gr aph with smal lest eigenvalue λ min ( h ) . If λ min ( h ) < − 2 , then λ min ( h ) ≤ − 1 − √ 2 . Pr o of. This follo ws immediately from Theorem 6.2 . No w w e prov e Theorem 1.6 . Pro of of Theorem 1.6 . F or λ ∈ ( − 2 , − 1], let d ′ λ = f ( λ ), where f ( λ ) is as defined in Theorem 1.5 , then ( G, σ ) is switc hing equiv alent to a p ositiv e clique and λ min ( G, σ ) = − 1. Assume that ( G, σ ) is obtained from ( K | V ( G ) | , +) by switc hing on U for some U ∈ V ( G ). Define the diagonal matrix D as D xx =  − 1 , if x ∈ U 1 , otherwise , then D = D − 1 = D T . Note that A ( G, σ ) = D ( A ( K | V ( G ) | , +)) D = D ( J − I ) D , where J is the all-ones matrix. Let N b e the matrix with order | V ( G ) | and with 1 on the first ro w and 0 on the rest, then J = N T N . Thus, A ( G, σ ) + I = D J D = D N T N D = ( N D ) T N D , and ( G, σ ) is 1-integrable. No w w e ma y assume that − 1 − √ 2 < λ ≤ − 2 and λ min ( G, σ ) ≤ − 2. T o obtain d ′ λ , we need to lo ok at the set F ( − 2). F or any f ∈ F ( − 2), λ min ( f ) ≤ − 1 − √ 2 , by Corollary 6.3 . Th us, there exists a p ositiv e in teger p f suc h that λ min ( G ( f , p f )) < λ , by Theorem 4.2 . Since F ( − 2) is finite, we may denote v f = max {| V f ( f ) | f ∈ F ( − 2) } , v s = max {| V s ( f ) | f ∈ F ( − 2) } and p = max { p f | f ∈ F ( − 2) } . Let m λ = min { m | λ min ( e K (0) 2 m ) < λ, λ min ( e K ( − ) 2 m ) < λ } and q = q ( m λ , v f , v s , p ) b e the in teger as defined in Prop osition 4.12 . Let n = max { q , q λ } and d ′ λ = max { d ( λ, n ) , 120 } , where q λ and d ( λ, n ) are as defined in Theorem 5.2 . Let g = g ( G, σ, m λ , n ) b e an associated Hoffman signed graph of ( G, σ ) with resp ect to the pair ( m λ , n ). Since λ min ( G, σ ) ≥ λ , λ min ( e K (0) 2 m λ ) < λ , λ min ( e K ( − ) 2 m λ ) < λ and λ min ( G ( f , p )) ≤ λ min ( G ( f , p f )) < λ , then ( G, σ ) is { e K (0) 2 m λ , e K ( − ) 2 m λ } -switc hing-free and ( G, σ ) do es not contain G ( f , p ) as an induced subgraph for an y f ∈ F ( − 2). 18 Th us, b y Prop osition 4.12 , g do es not contain an y Hoffman signed graph in F ( − 2) as an induced subgraph. As d ′ λ ≥ d ( λ, n ), Theorem 5.2 (i) implies that eac h v ertex of ( G, σ ) has at least one fat neighbor, consequently , g is fat. No w we show that λ min ( g ) ≥ − 2. Supp ose, for a con tradiction, λ min ( g ) < − 2, then g con tains a minimal fat Hoffman signed induced subgraph with smallest eigenv alue less than − 2. This contradicts that g do es not contain an y member of F ( − 2). It follows that λ min ( G, σ ) = − 2, as by Lemma 4.7 . Since ( G, σ ) con tains at least 1 + d λ ≥ 121 v ertices, signed graph ( G, σ ) is 1-in tegrable, b y Theorem 1.8 . This completes the proof. Ac kno wledgemen ts J.H. Ko olen is partially supported b y the National Natural Science F oundation of China (No. 12471335), and the Anhui Initiative in Quantum Information T echnologies (No. AHY150000). Q. Y ang is supp orted b y the National Natural Science F oundation of China (No. 12401460). Financial disclosure None rep orted. Conflict of in terest The authors declare no potential conflict of interests. References [1] I. Balla. Equiangular lines via matrix pro jection. A dv. Math. , 482:110620, 2025. [2] E. Bannai, E. Bannai, and D. Stanton. An upp er bound for the cardinalit y of an s -distance subset in real Euclidean space. II. Combinatoric a , 3(2):147–152, 1983. [3] F. Belardo, S. M. Cioab˘ a, J. H. 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