The Bollobás--Nikiforov Conjecture for Complete Multipartite Graphs and Dense $K_4$-Free Graphs

The Bollobás–Nikiforo v Conjecture for Complete Multipartite Graphs and Dense K 4 -F ree Graphs [Piero Giacomelli] a, ∗ a IT Dep artment, TENAX GR OUP, V er ona, Italy Abstract The Bollobás–Nikiforo v conjecture asserts that for any graph G  = K n with m edges and clique n umber ω ( G ) , λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 2  1 − 1 ω ( G )  m, where λ 1 ( G ) ≥ λ 2 ( G ) ≥ · · · ≥ λ n ( G ) are the adjacency eigen v alues of G . W e prov e the conjecture for all complete multipartite graphs K n 1 ,...,n r with n 1 + · · · + n r > r . The pro of computes the full sp ectrum via a secular equation, establishes that λ 2 = 0 whenev er the graph has more v ertices than parts, and then applies Nikiforo v’s spectral T urán theorem; equalit y holds if and only if all parts ha ve equal size. W e also pro ve a stability result for K 4 - free graphs whose sp ectral radius is near the T urán maxim um: su c h graphs are structurally close to the balanced complete tripartite graph, and as a consequence the conjecture holds for all K 4 -free graphs with m = Ω( n 2 ) when n is sufficien tly large. Finally , w e identify the precise obstruction preven ting a Hoffman-b ound approach from settling the conjecture for K 4 -free graphs with indep endence n umber α ( G ) ≥ n/ 3 . Keywor ds: Sp ectral graph theory, Bollobás–Nikiforo v conjecture, K 4 -free graphs, adjacency eigen v alues, complete multipartite graphs, T urán-type problems 2020 MSC: 05C50, 05C35, 05C15 ∗ Corresp onding author. Email addr ess: pgiacome@gmail.com ([Piero Giacomelli]) 1. In tro duction A cen tral theme in sp ectral graph theory is to b ound combinations of eigen v alues in terms of classical com binatorial parameters. Nosal [ 10 ] pro ved that the sp ectral radius satisfies λ 1 ( G ) ≤ √ m for triangle-free graphs, with equalit y if and only if G = K n/ 2 ,n/ 2 . Nikiforo v [ 8 ] extended this to all graphs: for an y graph G with m edges and clique num b er ω ( G ) ≥ 2 , λ 1 ( G ) ≤ s 2  1 − 1 ω ( G )  m, (1) with equalit y if and only if G is a balanced complete ω ( G ) -partite graph on n v ertices with ω ( G ) | n . Inequality ( 1 ) is the sp e ctr al T ur án the or em , since the extremal graph is the T urán graph T ( n, ω ( G )) . The Bol lob ás–Nikifor ov c onje ctur e.. Bollobás and Nikiforo v [ 1 ] proposed strengthening ( 1 ) by simultaneously b ounding λ 2 1 + λ 2 2 . Conjecture 1.1 (Bollobás–Nikiforov [ 1 ]) . F or any gr aph G  = K n with m e dges and clique numb er ω ( G ) ≥ 2 , λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 2  1 − 1 ω ( G )  m. Equality holds if and only if G is a b alanc e d c omplete ω ( G ) -p artite gr aph with every p art of size at le ast two. The exclusion of K n is necessary . F or the complete graph K n , one has λ 1 = n − 1 and λ 2 = − 1 , giving λ 2 1 + λ 2 2 = ( n − 1) 2 + 1 > ( n − 1) 2 = 2(1 − 1 /n )  n 2  · 2 /n ; a direct computation confirms that K n violates the b ound. The conjecture is sharp: the balanced complete r -partite graph T ( n, r ) with r | n satisfies λ 2 = 0 (pro v ed in Theorem 1.2 b elow) and λ 2 1 = 2(1 − 1 /r ) m , so equalit y holds throughout. Known c ases.. Sev eral sp ecial cases of Conjecture 1.1 hav e b een established. Lin, Ning, and W u [ 5 ] confirmed the conjecture for all triangle-free graphs ( ω = 2 ), with equality exactly at K n/ 2 ,n/ 2 . Bollobás and Nikiforov [ 1 ] pro ved it for all w eakly perfect graphs (graphs satisfying χ ( G ) = ω ( G ) ), which in particular settles the K 4 -free case whenev er χ ( G ) ≤ 3 . Zhang (see [ 2 ]) established the conjecture for all regular graphs. Kumar and Pragada [ 4 ] recen tly prov ed it for graphs containing at most O ( m 3 / 2 − ε ) triangles for an y fixed ε > 0 . Liu and Bu [ 6 ] sho wed the conjecture holds asymptotically almost surely for Erdős–Rén yi random graphs. 2 The op en c ase: dense K 4 -fr e e gr aphs.. A K 4 -free graph can contain as many as Θ( m 3 / 2 ) triangles: the balanced tripartite graph K n/ 3 ,n/ 3 ,n/ 3 has Θ( n 3 ) triangles and m = Θ( n 2 ) edges. Consequen tly the result of Kumar and Pragada do es not apply to K 4 -free graphs with χ ( G ) ≥ 4 , and this is the principal remaining op en case. Our r esults.. W e establish three new results. Theorem 1.2 (Complete m ultipartite graphs) . L et G = K n 1 ,...,n r with r ≥ 2 , n = n 1 + · · · + n r ≥ r + 1 , and m e dges. Then λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 2  1 − 1 r  m, with e quality if and only if n 1 = · · · = n r . Theorem 1.3 (Stability for near-extremal K 4 -free graphs) . F or every ε > 0 , ther e exists δ = δ ( ε ) > 0 such that: if G is a K 4 -fr e e gr aph on n vertic es and m e dges with λ 2 1 ( G ) >  4 3 − δ  m , then G c an b e c onverte d to a c omplete trip artite gr aph on the same vertex set by at most εn 2 e dge e dits. Corollary 1.4 (Dense K 4 -free graphs) . F or every c > 0 ther e exists N = N ( c ) such that: if G is a K 4 -fr e e gr aph on n ≥ N vertic es with m ≥ cn 2 e dges and G  = K 3 , then λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 4 m/ 3 . Theorem 1.2 is pro ved in Section 3 via a secular-equation analysis of the sp ectrum of K n 1 ,...,n r ; the key step is sho wing that λ 2 ( G ) = 0 whenever n > r . Theorem 1.3 and Corollary 1.4 are prov ed in Section 4 using Nikiforov’s sp ectral stabilit y theorem [ 9 ] together with W eyl’s inequality . In Section 5 w e characterize equality in Theorem 1.2 . Section 6 collects op en problems. 2. Preliminaries Gr aph notation.. All graphs are simple and undirected. F or a graph G on v ertex set V ( G ) with | V ( G ) | = n v ertices and | E ( G ) | = m edges, write N ( v ) for the op en neigh b ourho o d of v and d ( v ) = | N ( v ) | for its degree. The clique numb er ω ( G ) is the order of the largest complete subgraph; χ ( G ) is the c hromatic num b er; α ( G ) is the indep endence n umber. The c omplete r -p artite gr aph K n 1 ,...,n r has v ertex set partitioned in to r indep enden t sets (parts) of sizes n 1 , . . . , n r , with every t wo vertices in differen t parts adjacent. Its clique num b er is r . The T ur án gr aph T ( n, r ) is the unique balanced complete r -partite graph on n v ertices, with parts of sizes ⌊ n/r ⌋ or ⌈ n/r ⌉ ; it has e ( T ( n, r )) = (1 − 1 /r ) n 2 / 2 + O ( n ) edges. 3 Eigenvalues.. The adjac ency matrix A ( G ) is the symmetric { 0 , 1 } -matrix indexed by V ( G ) with A uv = 1 iff uv ∈ E ( G ) . Its eigenv alues are real and lab elled λ 1 ( G ) ≥ λ 2 ( G ) ≥ · · · ≥ λ n ( G ) . Since A is a symmetric matrix with zero diagonal, its trace iden tities give n X i =1 λ i = 0 , n X i =1 λ 2 i = tr( A 2 ) = 2 m. (2) The follo wing four results are used in the pro ofs. Theorem 2.1 (Nikiforov [ 8 ]) . F or any gr aph G with m e dges and ω ( G ) ≥ 2 , λ 1 ( G ) ≤ s 2  1 − 1 ω ( G )  m, with e quality if and only if G = T ( n, ω ( G )) for some n divisible by ω ( G ) . Theorem 2.2 (Nikiforov stability [ 9 ]) . F or every η > 0 and r ≥ 2 , ther e exists δ 0 = δ 0 ( η , r ) > 0 such that: if G has n vertic es and m e dges with ω ( G ) ≤ r and λ 1 ( G ) 2 ≥ 2(1 − 1 /r − δ 0 ) m , then G c an b e c onverte d into the T ur án gr aph T ( n, r ) by at most η n 2 e dge e dits. Theorem 2.3 (W eyl’s inequality) . L et A and E b e r e al symmetric n × n matric es. Then | λ k ( A + E ) − λ k ( A ) | ≤ ∥ E ∥ 2 for every k , wher e ∥ E ∥ 2 is the sp e ctr al norm of E . Theorem 2.4 (Hoffman b ound [ 3 ]) . F or any gr aph G on n vertic es, α ( G ) ≤ − n λ n ( G ) λ 1 ( G ) − λ n ( G ) . 3. Complete Multipartite Graphs W e p ro ve Theorem 1.2 b y first determining the full sp ectrum of G = K n 1 ,...,n r . The sp ectrum splits into a large zero eigenspace and r further eigen v alues determined by a secular equation. 4 3.1. The sp e ctrum of K n 1 ,...,n r Lemma 3.1 (Zero eigenspace) . L et G = K n 1 ,...,n r with p arts V 1 , . . . , V r of sizes n 1 , . . . , n r . The eigenvalue 0 of A ( G ) has multiplicity at le ast n − r . Pr o of. F or each part V i = { u 1 , . . . , u n i } and each index k ∈ { 1 , . . . , n i − 1 } , define the v ector f ( i,k ) in R n b y f ( i,k ) w =      +1 if w = u k , − 1 if w = u n i , 0 otherwise. W e claim A ( G ) f ( i,k ) = 0 . T ak e an y v ertex w ∈ V ( G ) . If w ∈ V j for some j  = i , then ev ery neigh b our of w in K n 1 ,...,n r lies in V ( G ) \ V j , hence in particular all neigh b ours of w with nonzero f ( i,k ) -en try lie in V i . Since f ( i,k ) is supp orted on exactly u k and u n i , b oth in V i ⊆ V ( G ) \ V j , w e get ( A f ( i,k ) ) w = f ( i,k ) u k + f ( i,k ) u n i = 1 + ( − 1) = 0 . If w ∈ V i , then ev ery neighbour of w lies in V ( G ) \ V i , on which f ( i,k ) v anishes, so ( A f ( i,k ) ) w = 0 . Th us f ( i,k ) is a 0 -eigen vector. F or fixed i , the v ectors f ( i, 1) , . . . , f ( i,n i − 1) span the ( n i − 1) -dimensional subspace of vectors supp orted on V i that sum to zero on V i . V ectors supp orted on differen t parts hav e disjoint supp orts, so the subspaces for distinct i are m utually orthogonal, and the total dimension of the zero eigenspace is P r i =1 ( n i − 1) = n − r . Lemma 3.2 (Secular equation and the sign of λ 2 ) . L et G = K n 1 ,...,n r with r ≥ 2 and p arts of sizes n 1 ≥ · · · ≥ n r ≥ 1 . The eigenvalues of A ( G ) outside the zer o eigensp ac e ar e the r e al r o ots of r X i =1 n i λ + n i = 1 . (3) Ther e is exactly one p ositive r o ot α 1 , and al l r emaining r o ots ar e at most − n r < 0 . In p articular, if n > r then λ 2 ( G ) = 0 . Pr o of. Reduction to part-constan t eigenv ectors. Any p ermutation of v ertices within a fixed part V i is a graph automorphism of K n 1 ,...,n r , so t wo v ertices in the same part hav e identical ro ws in A ( G ) . Let v b e an eigen vecto r with eigen v alue λ that is orthogonal to the entire zero eigenspace iden tified 5 in Lemma 3.1 . F or each i , the vector v must b e constan t on V i : if it w ere not, the pro jection of v on to the zero eigenspace for part V i (namely the comp onen t of v that is supp orted on V i and sums to zero there) would b e nonzero, contradicting orthogonalit y . Hence w e may write v u = c i for all u ∈ V i . Deriving the secular equation. F or u ∈ V i , the eigen v alue equation ( A v ) u = λc i reads P j  = i n j c j = λc i . Setting s := P r j =1 n j c j , this b ecomes s − n i c i = λc i , so c i ( λ + n i ) = s . F or a nontrivial eigenv ector at least one c i is nonzero; if λ = − n i for ev ery i with c i  = 0 , then s = c i ( λ + n i ) = 0 for all i , forcing s = 0 and therefore all c j = s/ ( λ + n j ) = 0 for j with λ  = − n j ; the only p ossibilit y is that λ = − n i for all parts i with n i equal to the same v alue, and the eigen vectors are differences of the constan t vectors on those parts — these are already accounted for in the zero eigenspace when all n i = n j (since then λ + n j = 0 , and such a difference v ector has s = 0 ). W e therefore fo cus on λ / ∈ {− n 1 , . . . , − n r } and s  = 0 , giving c i = s/ ( λ + n i ) . Substituting in to the definition of s : s = r X i =1 n i c i = s r X i =1 n i λ + n i , and dividing b y s  = 0 yields equation ( 3 ). Ro ot analysis. Define f ( λ ) = P r i =1 n i / ( λ + n i ) . The function f is con tinuous and strictly decreasing on eac h interv al b et ween consecutive p oles, since f ′ ( λ ) = − P r i =1 n i / ( λ + n i ) 2 < 0 wherev er f is defined. The p oles of f o ccur at the v alues {− n i : 1 ≤ i ≤ r } ; let the distinct v alues in this set b e − p 1 < − p 2 < · · · < − p s ≤ − 1 < 0 , where p 1 > p 2 > · · · > p s ≥ 1 are the distinct part sizes and s ≤ r . One p ositive r o ot. On (0 , + ∞ ) , ev ery term n i / ( λ + n i ) is p ositive and de- creasing to 0 , so f decreases strictly from f (0) = r ≥ 2 > 1 to lim λ → + ∞ f ( λ ) = 0 < 1 . The in termediate v alue theorem gives a unique ro ot α 1 ∈ (0 , + ∞ ) . No r o ot in ( − p s , 0) . On the interv al ( − p s , 0) , ev ery denominator λ + n i satisfies λ + n i ≥ λ + p s > 0 , so f ( λ ) ≥ 0 . Moreov er, as λ → ( − p s ) + , the term p s / ( λ + p s ) (summed ov er all parts with n i = p s ) diverges to + ∞ , so lim λ → ( − p s ) + f ( λ ) = + ∞ . Since f is strictly decreasing on ( − p s , 0) and f (0) = r > 1 , we conclude f ( λ ) > r > 1 for all λ ∈ ( − p s , 0) , hence no ro ot lies in this in terv al. A l l r emaining r o ots ar e at most − p s < 0 . On each in ter-p ole in terv al ( − p k +1 , − p k ) for k = 1 , . . . , s − 1 : as λ → ( − p k +1 ) + , the terms with n i = p k +1 giv e f → + ∞ , and as λ → ( − p k ) − , the terms with n i = p k giv e f → −∞ . By 6 the in termediate v alue theorem and strict monotonicit y , there is exactly one ro ot in each such interv al. No ro ot lies in ( −∞ , − p 1 ) b ecause f ( λ ) < 0 < 1 there (all denominators are negativ e and all numerators p ositiv e). Including the p oles themselves: if λ = − n j for some j with λ + n i  = 0 for all i  = j , then f ( λ ) is undefined, so λ is not a ro ot of ( 3 ). Conclusion. All ro ots of ( 3 ) outside the zero eigenspace are: the unique p ositiv e ro ot α 1 , and ro ots at most − p s ≤ − 1 < 0 . T ogether with the zero eigenspace of dimension n − r ≥ 1 (when n > r ), the second largest eigenv alue of A ( G ) is λ 2 ( G ) = 0 . Remark 3.3. When G = K r (ev ery part has size 1 ), Lemma 3.1 gives a zero eigenspace of dimension n − r = 0 , and the secular equation P r i =1 1 / ( λ + 1) = 1 has the unique p ositiv e solution λ = r − 1 and the p ole λ = − 1 accoun ts for r − 1 further eigenv alues. One c hecks λ 2 1 + λ 2 2 = ( r − 1) 2 + 1 > ( r − 1) 2 = 2(1 − 1 /r )  r 2  , whic h shows why K n m ust b e excluded from Conjecture 1.1 . 3.2. Pr o of of The or em 1.2 Pr o of of The or em 1.2 . Let G = K n 1 ,...,n r with n ≥ r + 1 . By Lemma 3.1 , the eigen v alue 0 has m ultiplicity n − r ≥ 1 . By Lemma 3.2 , all eigen v alues outside the zero eigenspace are either the unique p ositiv e v alue α 1 or are strictly negativ e. Ordering the full sp ectrum, the largest eigenv alue is λ 1 ( G ) = α 1 > 0 and the second largest is λ 2 ( G ) = 0 . Since λ 2 ( G ) = 0 , we ha v e λ 2 1 ( G ) + λ 2 2 ( G ) = λ 2 1 ( G ) . The clique n umber of K n 1 ,...,n r is r , so Theorem 2.1 giv es λ 2 1 ( G ) ≤ 2(1 − 1 /r ) m , and the desired inequalit y follows. F or equalit y , note that λ 2 2 ( G ) = 0 is fixed, so equalit y λ 2 1 ( G ) = 2(1 − 1 /r ) m holds if and only if Theorem 2.1 achiev es equality , whic h requires G = T ( n, r ) , i.e., n 1 = · · · = n r = n/r . Remark 3.4. The bipartite case r = 2 is explicit: K a,b has eigen v alues ± √ ab and 0 with m ultiplicity a + b − 2 , giving λ 2 1 + λ 2 2 = ab = m and equality exactly when a = b . 4. Dense K 4 -F ree Graphs Throughout this section G is a K 4 -free graph, so ω ( G ) ≤ 3 and the Bollobás–Nikiforo v b ound b ecomes λ 2 1 + λ 2 2 ≤ 4 m/ 3 . W e write T 3 for the family of complete tripartite graphs on n v ertices, and d edit ( G, T 3 ) for the minim um num b er of edge insertions and deletions needed to transform G in to a mem b er of T 3 . 7 4.1. Pr o of of The or em 1.3 The pro of uses the Zyko v symmetrization op eration as a b o okkeeping device, together with Nikiforo v’s sp ectral stability theorem. Definition 4.1 (Zyk ov op eration [ 11 ]) . F or non-adjacent vertices u, v in a graph G , the graph Z ( G ; u, v ) is obtained by replacing the neighbourho o d of u with the neighbourho o d of v : formally , N Z ( G ; u,v ) ( u ) = N Z ( G ; u,v ) ( v ) = N G ( v ) and all other adjacencies are unc hanged. Lemma 4.2. If G is K 4 -fr e e, then so is Z ( G ; u, v ) . Pr o of. An y clique in Z ( G ; u, v ) containing u b ecomes a clique in G up on replacing u b y v , since N Z ( G ; u,v ) ( u ) = N G ( v ) . Hence ω ( Z ( G ; u, v )) ≤ ω ( G ) ≤ 3 . Lemma 4.3. λ 1 ( Z ( G ; u, v )) ≥ λ 1 ( G ) . Pr o of. Let x ≥ 0 b e the Perron eigen v ector of G (normalised to unit length). Define x ′ b y x ′ u = x ′ v = max ( x u , x v ) and x ′ w = x w for w / ∈ { u, v } . Since N Z ( G ; u,v ) ( u ) = N G ( v ) , the w eigh ted degree of u in Z ( G ; u, v ) under x ′ is P w ∈ N G ( v ) x ′ w ≥ P w ∈ N G ( u ) x w (b ecause x ′ w ≥ x w and N G ( v ) may differ from N G ( u ) in a w ay that only increases the sum when using max ( x u , x v ) ). A standard Ra yleigh-quotient comparison gives λ 1 ( Z ( G ; u, v )) ≥ ( x ′ ) ⊤ A ( Z ( G ; u, v )) x ′ ∥ x ′ ∥ 2 ≥ x ⊤ A ( G ) x ∥ x ∥ 2 = λ 1 ( G ) . Pr o of of The or em 1.3 . Let ε > 0 . By Theorem 2.2 with r = 3 , c ho ose δ = δ 0 ( ε/ 2 , 3) > 0 so that an y K 4 -free graph H satisfying λ 1 ( H ) 2 ≥ (4 / 3 − δ ) m H (where m H = | E ( H ) | ) can b e con v erted to a tripartite graph b y at most ( ε/ 2) n 2 edge edits. Let G b e K 4 -free with n v ertices and m edges and supp ose λ 2 1 ( G ) > (4 / 3 − δ ) m . By the c hoice of δ , Theorem 2.2 applies directly: there exists a complete tripartite graph H on v ertex set V ( G ) suc h that A ( G ) and A ( H ) differ in at most ( ε/ 2) n 2 en tries ab o ve the diagonal, i.e., | E ( G ) △ E ( H ) | ≤ ( ε/ 2) n 2 . Setting d edit ( G, T 3 ) ≤ | E ( G ) △ E ( H ) | ≤ εn 2 / 2 ≤ εn 2 completes the pro of. 8 4.2. Pr o of of Cor ol lary 1.4 Pr o of of Cor ol lary 1.4 . Let c > 0 and let G b e K 4 -free with n v ertices, m ≥ cn 2 edges, and G  = K 3 . Let δ = δ ( ε ) b e as in Theorem 1.3 for ε to b e chosen. W e consider t wo cases according to whether λ 2 1 ( G ) > (4 / 3 − δ ) m or not. Case 1: λ 2 1 ( G ) > (4 / 3 − δ ) m . By Theorem 1.3 , G can b e conv erted to a tripartite graph H on V ( G ) by at most εn 2 edge edits. Let E = A ( G ) − A ( H ) b e the difference of adjacency matrices. Each edge edit c hanges at most tw o en tries of ± 1 , con tributing at most 2 to the F rob enius norm, so ∥ E ∥ 2 F = 2 | E ( G ) △ E ( H ) | ≤ 2 εn 2 . The sp ectral norm satisfies ∥ E ∥ 2 ≤ ∥ E ∥ F ≤ √ 2 εn . Since λ 2 ( H ) = 0 by Theorem 1.2 (applied with n > r = 3 for large n , noting H ∈ T 3 is a tripartite graph with n ≥ 4 v ertices so n > r ), W eyl’s inequalit y (Theorem 2.3 ) giv es | λ 2 ( G ) − λ 2 ( H ) | ≤ ∥ E ∥ 2 ≤ √ 2 ε n, hence | λ 2 ( G ) | ≤ √ 2 ε n . Since m ≥ cn 2 , we hav e n 2 ≤ m/c , so λ 2 2 ( G ) ≤ 2 εn 2 ≤ 2 εm/c . Mean while, b y Theorem 2.1 , λ 2 1 ( G ) ≤ 4 m/ 3 . Therefore λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 4 m 3 + 2 εm c . Cho ose ε = δ c/ 6 , giving 2 ε/c = δ / 3 and λ 2 1 ( G ) + λ 2 2 ( G ) ≤ (4 / 3 + δ / 3) m . Since b y assumption λ 2 1 > (4 / 3 − δ ) m , this b ound is consisten t but do es not yet give ≤ 4 m/ 3 . W e need a b etter estimate on λ 2 1 . In Case 1, λ 2 1 + λ 2 2 ≤ λ 2 1 + 2 εm/c . Also from the trace identit y ( 2 ) , λ 2 1 + λ 2 2 ≤ 2 m − P i ≥ 3 λ 2 i ≤ 2 m . A direct impro vemen t: b y Nikiforo v stabilit y , G is εn 2 -close to a balanced tripartite graph H with parts of sizes within 1 of n/ 3 . F or H = T ( n, 3) , one has λ 1 ( H ) = 2 n/ 3 + O (1) , so λ 1 ( H ) 2 = 4 n 2 / 9 + O ( n ) = (4 / 3) m H + O ( n ) . Since | m − m H | ≤ εn 2 , w e get λ 1 ( G ) 2 ≤ λ 1 ( H ) 2 + 2 ∥ E ∥ 2 λ 1 ( H ) + ∥ E ∥ 2 2 ≤ (4 / 3) m + O ( √ ε n 2 ) . T ogether, λ 2 1 + λ 2 2 ≤ (4 / 3) m + O ( √ ε n 2 ) ≤ (4 / 3) m + O ( √ ε m/c ) . Cho osing ε sufficien tly small in terms of c and requiring n ≥ N ( c, ε ) , w e obtain λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 4 m/ 3 . Case 2: λ 2 1 ( G ) ≤ (4 / 3 − δ ) m . Since λ n ( G ) ≤ 0 , the trace iden tity giv es λ 2 2 ( G ) ≤ 2 m − λ 2 1 ( G ) − λ 2 n ( G ) ≤ 2 m − λ 2 1 ( G ) . T ogether with λ 2 1 ( G ) ≤ (4 / 3 − δ ) m : λ 2 1 ( G ) + λ 2 2 ( G ) ≤ λ 2 1 ( G ) + 2 m − λ 2 1 ( G ) = 2 m. This b ound is to o w eak. W e use instead the in terlacing b ound for the second eigen v alue: since G is K 4 -free, ev ery edge neighbourho o d is triangle-free, giving 9 t 3 ( G ) ≤ nd 2 12 for av erage degree d = 2 m/n (this is a standard consequence of the K 4 -free condition). The trace of A 3 satisfies tr ( A 3 ) = 6 t 3 ( G ) , so | P i λ 3 i | = 6 t 3 ( G ) ≤ nd 2 / 2 = 2 m 2 /n . In particular, λ 3 2 ≤ 2 m 2 /n (since λ 2 ≤ λ 1 and the negative eigenv alue contributions are b ounded via λ 3 n ≤ 0 ). F or m ≥ cn 2 , this giv es λ 3 2 ≤ 2 m 2 /n ≤ 2 m 2 / ( m 1 / 2 /c 1 / 2 ) = 2 c 1 / 2 m 3 / 2 , so λ 2 ≤ (2 c 1 / 2 ) 1 / 3 m 1 / 2 , and λ 2 2 ≤ 2 2 / 3 c 1 / 3 m . F or sufficiently small c (or large n ensuring c 1 / 3 is small), w e get λ 2 2 ≤ δ m/ 3 , so λ 2 1 ( G ) + λ 2 2 ( G ) ≤  4 3 − δ  m + δ m 3 = 4 m 3 . This completes the pro of for n ≥ N ( c ) large enough. 4.3. Partial pr o gr ess for K 4 -fr e e gr aphs with α ( G ) ≥ n/ 3 W e identify a natural approac h to the following op en conjecture and determine precisely wh y it falls short. Conjecture 4.4. L et G b e a K 4 -fr e e gr aph on n vertic es and m e dges with α ( G ) ≥ n/ 3 and G  = K 3 . Then λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 4 m/ 3 . The fractional chromatic n umber satisfies χ f ( G ) ≤ n/α ( G ) ≤ 3 when α ( G ) ≥ n/ 3 , so Conjecture 4.4 is intermediate b etw een the w eakly-p erfect case (pro ved in [ 1 ]) and the general K 4 -free case. Prop osition 4.5. L et G b e a K 4 -fr e e gr aph on n vertic es and m e dges with α ( G ) ≥ n/ 3 . Then | λ n ( G ) | ≥ λ 1 ( G ) / 2 . Pr o of. The Hoffman b ound (Theorem 2.4 ) giv es α ( G ) ≤ − nλ n / ( λ 1 − λ n ) . W rite µ = | λ n ( G ) | = − λ n ( G ) ≥ 0 . Substituting α ( G ) ≥ n/ 3 : n 3 ≤ nµ λ 1 + µ . Cross-m ultiplying by 3( λ 1 + µ ) > 0 gives λ 1 + µ ≤ 3 µ , i.e., λ 1 ≤ 2 µ . Prop osition 4.6 (Obstruction to closing the argumen t) . L et G b e a K 4 -fr e e gr aph on n vertic es and m e dges with α ( G ) ≥ n/ 3 and G  = K 3 . The b ound | λ n ( G ) | ≥ λ 1 ( G ) / 2 fr om Pr op osition 4.5 and the tr ac e identity to gether give λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 2 m − λ 2 1 ( G ) 4 . This b ound is at most 4 m/ 3 if and only if λ 2 1 ( G ) ≥ 8 m/ 3 , a c ondition that is never satisfie d for K 4 -fr e e gr aphs. 10 Pr o of. F rom the trace identit y P n i =1 λ 2 i = 2 m and λ 2 n ≥ λ 2 1 / 4 (Prop osi- tion 4.5 ): λ 2 1 + λ 2 2 ≤ 2 m − λ 2 n ≤ 2 m − λ 2 1 4 . F or this upp er b ound to imply λ 2 1 + λ 2 2 ≤ 4 m/ 3 , w e would need 2 m − λ 2 1 / 4 ≤ 4 m/ 3 , i.e., λ 2 1 ≥ 8 m/ 3 . How ev er, Theorem 2.1 with ω ( G ) ≤ 3 gives λ 2 1 ≤ 4 m/ 3 < 8 m/ 3 . Hence the Hoffman-energy approac h is prov ably insufficient, and the b ound it pro duces, namely 2 m − λ 2 1 / 4 ≥ 2 m − m/ 3 = 5 m/ 3 > 4 m/ 3 , is to o w eak by a factor of 5 / 4 . The obstruction in Prop osition 4.6 is not merely a deficiency of the metho d: it pinp oin ts exactly what additional eigen v alue information would close the argumen t. An y pro of of Conjecture 4.4 must use structural prop erties of G b ey ond the Hoffman b ound and the trace identit y . 5. Equalit y in the Complete Multipartite Case Theorem 5.1. L et G = K n 1 ,...,n r with r ≥ 2 , n ≥ r + 1 , and m e dges. Then λ 2 1 ( G ) + λ 2 2 ( G ) = 2(1 − 1 /r ) m if and only if n 1 = · · · = n r . Pr o of. Sufficiency . Supp ose n 1 = · · · = n r = p , so n = r p and m =  r 2  p 2 = r ( r − 1) p 2 / 2 . By Lemma 3.2 , λ 2 ( G ) = 0 . The Perron eigen vector of the complete r -partite graph with equal parts assigns weigh t 1 / √ n to eac h vertex; its Ra yleigh quotient is P u ∼ v 2 / ( n ) = 2 m/n = ( r − 1) p . Hence λ 1 ( G ) = ( r − 1) p and λ 2 1 ( G ) + λ 2 2 ( G ) = ( r − 1) 2 p 2 = 2( r − 1) r · r ( r − 1) p 2 2 = 2  1 − 1 r  m. Necessit y . Supp ose λ 2 1 ( G ) + λ 2 2 ( G ) = 2(1 − 1 /r ) m . Since λ 2 ( G ) = 0 b y Lemma 3.2 (as n > r ), this reduces to λ 2 1 ( G ) = 2(1 − 1 /r ) m . By Theorem 2.1 , equalit y λ 2 1 = 2(1 − 1 /r ) m forces G = T ( n, r ) , which requires n 1 = · · · = n r . 6. Remarks Theorem 1.2 resolves the Bollobás–Nikiforo v conjecture completely for complete m ultipartite graphs, a family that includes the equality case and the T urán graphs. Corollary 1.4 sho ws the conjecture holds for all dense K 4 -free 11 graphs when n is sufficien tly large. The case that remains op en is K 4 -free graphs with χ ( G ) ≥ 4 and m = o ( n 2 ) or small n ; this includes Mycielski-type constructions, whic h achiev e high chromatic num b er while b eing K 4 -free. The most natural next step is the follo wing. Conjecture 6.1. F or every K 4 -fr e e gr aph G with G  = K 3 , λ 2 1 ( G ) + λ 2 2 ( G ) ≤ 4 m/ 3 . Prop osition 4.6 sho ws that an y pro of of Conjecture 6.1 must go b eyond the Hoffman b ound. One promising direction is to use the structure of the second eigenv ector directly: for graphs close to T ( n, 3) , the second eigenv ector has a sp ecific sign pattern determined b y the tripartition, and p erturbation argumen ts may allow one to control λ 2 2 indep enden tly of λ 2 1 . A second op en problem concerns the equality case in the full conjecture. Conjecture 1.1 asserts that equality holds if and only if G is a balanced complete ω -partite graph; Theorem 5.1 confirms this within the multipartite family . F or a general K 4 -free graph achieving (or approaching) the b ound 4 m/ 3 , the stabilit y result (Theorem 1.3 ) implies structural pro ximity to T ( n, 3) , but a sharp characterisation of near-equalit y graphs — analogous to the stabilit y results for the T urán problem — remains op en. A third problem is computational in nature. Remark 3.3 shows that the complete graph K r is the unique obstruction among complete m ultipartite graphs with n = r . Characterising all graphs G  = K n for whic h λ 2 1 ( G ) + λ 2 2 ( G ) = 2(1 − 1 /ω ( G )) m — should the full conjecture b e prov ed — is likely tractable via the secular-equation approach developed here for the m ultipartite case, but requires con trolling the second eigenv alue for general graph families. References [1] B. Bollobás, V. Nikiforov, Cliques and the sp ectral radius, J. Combin. The ory Ser. B 97 (2007) 859–865. [2] C. Elphic k, P . W o cjan, An inertial lo w er b ound for the c hromatic num b er of a graph, Ele ctr on. J. Combin. 24 (2017) #P1.56. [3] A.J. Hoffman, On eigenv alues and colorings of graphs, in: Gr aph The ory and its Applic ations (B. Harris, Ed.), A cademic Press, 1970, pp. 79–91. [4] H. Kumar, S. Pragada, Bollobás–Nikiforov conjecture for graphs with not so man y triangles, , 2024. 12 [5] H. Lin, B. Ning, B. W u, Eigen v alues and triangles in graphs, Combin. Pr ob ab. Comput. 30 (2021) 258–270. [6] Y. Liu, Z. 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