Localization of the clique spectral version of Zykov's theorem

Lo calization of the clique sp ectral v ersion of Zyk o v’s theorem Chang jiang Bu a, ∗ , Jueru Liu a , Haotian Zeng a a Scho ol of Mathematic al Scienc es, Harbin Engine ering University, Harbin 150001, PR China Abstract Zyk ov’s theorem sho ws that r -partite T ur´ an graph uniquely has the maxim um num- b er of K t among all n -v ertex K r +1 -free graphs for 2 ≤ t ≤ r . The clique tensor is a high-order extension of the adjacency matrix of a graph. Y u and Peng [26] gav e a sp ectral version of the Zyk ov’s theorem via clique tensor. In this pap er, we giv e some upp er b ounds on the sp ectral radius of the clique tensor of a graph, whic h can b e view ed as the lo calizations of the sp ectral v ersion of Zyko v’s theorem. Keywor ds: tensor, sp ectral radius, clique AMS classific ation (2020): 05C35, 15A42, 15A69 1. In tro duction The graphs considered throughout this pap er are all simple and undirected. F or a graph G , if an induced subgraph of a subset of V ( G ) is a complete graph, then the subset is called a clique . The clique numb er of G is the n umber of vertices of a largest clique in G , denoted by ω ( G ). A clique is called a t -clique if it has t v ertices. Let C t ( G ) b e the set of all t -cliques in G . Let ρ ( G ) denote the sp ectral radius of G . In 2002, Nikiforo v [19] gav e an upp er b ound on the sp ectral radius of graphs. Theorem 1.1. [19] L et G b e an n -vertex gr aph with clique numb er ω . Then ρ ( G ) ≤ s 2 | E ( G ) |  1 − 1 w  . Equality holds if and only if G is a c omplete bip artite gr aph for r = 2 , or a c omplete r -p artite gr aph for r ≥ 3 and r divides n (p ossibly with some isolate d vertic es). ∗ Corresp onding author Email addr ess: buchangjiang@hrbeu.edu.cn (Changjiang Bu) 1 In fact, the ab o ve conclusion implies the concise T ur´ an’s theorem. Theorem 1.2. [22] L et G b e an n -vertex K r +1 -fr e e gr aph (i.e., c ontaining no c opy of the c omplete gr aph K r +1 ). Then | E ( G ) | ≤  1 − 1 r  n 2 2 . Equality holds if and only if r divides n and G is a c omplete r e gular r -p artite gr aph. The T ur´ an num b er for a graph F is the maxim um n umber of edges in an n -vertex F -free graph. Some results on T ur´ an problem and spectral T ur´ an problem can b e referred to [2, 6, 23, 24]. Recen tly , Brada ˇ c [5] and Malec and T ompkins [18] ga v e a lo calized version of concise T ur´ an’s theorem. F or an edge e ∈ E ( G ), let α ( e ) be the order of the largest clique in G containing e . Theorem 1.3. [5, 18] L et G b e an n -vertex gr aph. Then X e ∈ E ( G ) α ( e ) α ( e ) − 1 ≤ n 2 2 . Equality holds if and only if G is a c omplete multi-p artite gr aph with vertex classes of e qual size. Liu et al. [15] ga ve an upp er b ound on the sp ectral radius of a graph in terms of the order of the largest clique con taining eac h edge, whic h is a local version of sp ectral T ur´ an’s theorem. And some other results on lo calization of spectral T ur´ an’s theorem were given in [14]. Theorem 1.4. [15] L et G b e a gr aph with clique numb er ω . Then ρ ( G ) ≤ v u u t 2 X e ∈ E ( G ) α ( e ) − 1 α ( e ) . Equality holds if and only if G is a c omplete bip artite gr aph for ω = 2 , or a c omplete ω -p artite gr aph for ω ≥ 3 and ω divides n (p ossibly with some isolate d vertic es). As an edge can b e view ed as an induced subgraph of a 2-clique, the generalized T ur´ an n umber ex( n, H , F ) studies the maxim um num b er of copies of subgraphs H in an n -v ertex F -free graph. The famous generalized T ur´ an result standing on 2 its o wn is the complete determination of ex( n, K t , K r +1 ) b y Zyko v [27] and Erd˝ os [7]. Subsequen tly , Alon and Shikhelman [3] studied the function ex( n, H , F ), some results on the generalized T ur´ an n umbers can be referred to [8, 9, 16, 17]. In 2024, Kirsc h and Nir [10] prop osed a localized approac h to generalized T ur´ an problems and ga ve a lo calized v ersion of Zyk ov’s theorem b y assigning weigh ts to cliques of an y size. The sp ectral T ur´ an problems hav e attracted considerable atten tion, but there are few studies on sp ectral versions of generalized T ur´ an num b ers. In 2023, Liu and Bu [12] prop osed the clique tensor of a graph and gav e a gen- eralization of the sp ectral Mantel’s theorem. Recently , some results on the spectral v ersion of the generalized T ur´ an num b er were giv en via clique tensor [13, 25, 26]. In 2025, Y u and Peng [26] ga ve a sp ectral v ersion of Zyko v’s theorem, which sho ws that the complete regular r -partite graph attains the maximum t -clique spectral radius among all n -v ertex K r +1 -free graphs. In 2026, a tensor’s spectral bound on the clique num b er was given [13], which extends Nikiforo v’s theorem (Theorem 1.1) to clique tensors. Theorem 1.5. [13] L et G b e a gr aph with clique numb er ω . F or 2 ≤ t ≤ ω , ρ t ( G ) ≤ t ω  ω t  1 t | C t ( G ) | t − 1 t . Mor e over, if G is a c omplete r e gular ω -p artite gr aph for ω ≥ t ≥ 2 , then the e quality is achieve d in the ab ove ine quality. In this pap er, w e give some upper b ounds on the t -clique spectral radius of graphs in terms of the order of the largest clique con taining each clique or v ertex, whic h can b e viewed as the lo calizations of the sp ectral version of Zyk o v’s theorem. 2. Preliminaries In this section, some related definitions and lemmas are introduced. F or a p os- itiv e integer n , let [ n ] = { 1 , 2 , . . . , n } . A k -order n -dimensional complex tensor A = ( a i 1 i 2 ··· i k ) is a multi-dimensional array with n k en tries on complex num b er field C , where i 1 i 2 · · · i k ∈ [ n ] k . Denote the set of n -dimensional complex v ectors and the set of k -order n -dimensional complex tensors by C n and C [ k,n ] , resp ectiv ely . F or A = ( a i 1 i 2 ··· i k ) ∈ C [ k,n ] and x = ( x 1 , . . . , x n ) T ∈ C n , A x k − 1 is a v ector in C n whose 3 i -th comp onen t is  A x k − 1  i = n X i 2 ,...,i k =1 a ii 2 ··· i k x i 2 · · · x i k . A num b er λ ∈ C is called an eigenvalue of A if there exists a nonzero vector x ∈ C n suc h that A x k − 1 = λx [ k − 1] , where x [ k − 1] = ( x k − 1 1 , . . . , x k − 1 n ) T and x is called an eigenve ctor of A asso ciated with λ [11, 20]. The sp e ctr al r adius of A is the maxim um mo dulus of all eigenv alues of A , denoted b y ρ ( A ). A tensor A is termed symmetric if its en tries remain in v ariant under an y p er- m utation of their indices. F urthermore, if all entries of a tensor A are nonnegativ e, then A is referred to as a nonne gative tensor . Let R n + (resp. R n ++ ) be the set of all n -dimensional vectors with nonnegativ e (resp. p ositiv e) comp onents. Lemma 2.1. [21] L et A = ( a i 1 i 2 ··· i k ) b e a k -or der n -dimensional symmetric nonne g- ative tensor. The sp e ctr al r adius of A is e qual to max { n X i 1 ,i 2 ,...,i k =1 a i 1 i 2 ··· i k x i 1 x i 2 · · · x i k : n X i =1 x k i = 1 , ( x 1 , x 2 , . . . , x n ) T ∈ R n + } . F or an n -v ertex graph G and an in teger t (2 ≤ t ≤ ω ( G )), the t -clique tensor A ( G ) = ( a i 1 i 2 ··· i t ) is a t -order n -dimensional tensor, with entries [12] a i 1 i 2 ··· i t = ( 1 ( t − 1)! , if { i 1 , i 2 , . . . , i t } ∈ C t ( G ) , 0 , otherwise . Sp ecifically , the 2-clique tensor is the adjacency matrix of G . The sp ectral radius of A ( G ) is called the t -clique sp e ctr al r adius of G , denoted by ρ t ( G ). It is pro ved that | C t ( G ) | ≤ n t ρ t ( G ) and equalit y holds if the n umber of t -cliques containing each v ertex in V ( G ) is equal [12]. F or v ∈ V ( G ), let c t ( v ) b e the n um b er of t -cliques that con tain the v ertex i in G . Next, w e describ e the necessary and sufficient conditions for the equalit y to hold. Lemma 2.2. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . Then | C t ( G ) | ≤ n t ρ t ( G ) . 4 Equality holds if and only if the numb er of t -cliques c ontaining e ach vertex in V ( G ) is e qual. Pr o of. Without loss of generalit y , let V ( G ) = [ n ]. Let A ( G ) = ( a i 1 i 2 ··· i t ) b e the t -clique tensor of G . F or i ∈ [ n ], let c t ( i ) denote the n umber of t -cliques contain the v ertex i in G . Then n X i 2 ,...,i t =1 a ii 2 ··· i t = c t ( i ) , i ∈ [ n ] . Let x ∈ R n + b e a vector with entries x i = n − 1 t ( i ∈ [ n ]). F rom Lemma 2.1, w e kno w that ρ t ( G ) ≥ n X i 1 ,i 2 ,...,i t =1 a i 1 i 2 ··· i t x i 1 x i 2 · · · x i t = t · | C t ( G ) | n . (2.1) If | C t ( G ) | = n t ρ t ( G ), then the equality holds in Eq.(2.1). It follo ws that the all- one v ector 1 ∈ R n is the eigenv ector of A ( G ) asso ciated with ρ t ( G ), i.e., A ( G ) 1 t − 1 = ρ t ( G ) 1 [ t − 1] . So, w e ha ve ρ t ( G ) =  A ( G ) 1 t − 1  i = n X i 2 ,...,i t =1 a ii 2 ··· i t = c t ( i ) , i ∈ [ n ] , whic h implies that the num b er of t -cliques con taining each vertex in V ( G ) is equal. If the num b er of t -cliques con taining each vertex in V ( G ) is equal, then ρ t ( G ) = t ·| C t ( G ) | n [12], completing the pro of. F or a vector x = ( x 1 , x 2 , . . . , x n ) T ∈ R n and a set I ⊆ [ n ], denote the pro duct x I = Q i ∈ I x i . Given a graph G , for a t -clique I ∈ C t ( G ), let α ( I ) b e the order of the largest clique in G con taining I , where 2 ≤ t ≤ ω ( G ). F or tw o in tegers s and q with 1 ≤ s ≤ q , define the follo wing homogeneous p olynomials h s,G ( x ) = X J ∈ C s ( G ) x J and f s,q ,G ( x ) = X I ∈ C q ( G )  α ( I ) s  q s  α ( I ) q  − 1 x I . Lemma 2.3. [4] F or every x ∈ R n + , then f s,q ,G ( x ) ≤ h s,G ( x ) q s . 5 Mor e over, e quality holds for x ∈ R n ++ only when the sub gr aph of G induc e d on the set of vertic es that b elong to an s -clique is a c omplete l -p artite gr aph with p arts V 1 , . . . , V l , for some l ≥ q , and P v ∈ V i x v = P u ∈ V j x u for al l 1 ≤ i, j ≤ l . F or a v ector x ∈ R n , the supp ort of x , denoted b y supp( x ), is the set of all indices corresp onding to nonzero en tries in x . When s = 1 and x = ( x 1 , . . . , x n ) T ∈ R n + is a v ector with || x || 1 = x 1 + · · · + x n = 1, from Lemma 2.3, w e can get the following conclusion directly . Lemma 2.4. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . F or any ve ctor x ∈ R n + with || x || 1 = 1 , X I ∈ C t ( G ) ( α ( I )) t  α ( I ) t  − 1 x I ≤ 1 , Equality holds if and only if the induc e d sub gr aph of G on supp( x ) is a c omplete ω -p artite gr aph with p artition V 1 , V 2 , . . . , V ω satisfying P v ∈ V i x v = 1 ω for al l i ∈ [ ω ] . F or a graph G and a v ertex v ∈ V ( G ), let α ( v ) denote the order of the largest clique containing v in G . F or a t -clique I = { i 1 , i 2 , . . . , i t } in G , it is clearly α ( I ) ≤ min { α ( i 1 ) , α ( i 2 ) , . . . , α ( i t ) } . Hence, w e ha ve the following conclusion. Lemma 2.5. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . F or any ve ctor x ∈ R n + with || x || 1 = 1 , X I = { i 1 ,i 2 ,...,i t }∈ C t ( G ) 1 t t X j =1 ( α ( i j )) t  α ( i j ) t  − 1 ! x I ≤ 1 . Equality holds if and only if the induc e d sub gr aph of G on supp( x ) is a c omplete ω -p artite gr aph with p artition V 1 , V 2 , . . . , V ω satisfy P v ∈ V i x v = 1 ω for al l i ∈ [ ω ] . 3. Main results In this section, w e obtain some upp er b ounds on the t -clique sp ectral radius of a graph, whic h are expressed b y the order of the largest clique con taining each clique or v ertex and can be view ed as the lo calized v ersions of the sp ectral Zyk ov’s theorem. 6 Theorem 3.1. L et G b e a gr aph with t -clique sp e ctr al r adius ρ t ( G ) and clique num- b er ω and let 2 ≤ t ≤ ω . Then  ρ t ( G ) t  t ≤   X I ∈ C t ( G ) t − 1 s  α ( I ) t  ( α ( I )) − t   t − 1 . Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c ontaine d in t -cliques is a c omplete t -p artite gr aph for ω = t , or a c omplete r e gular ω -p artite gr aph for ω ≥ t + 1 (p ossibly with some isolate d vertic es). Pr o of. F or the t -clique tensor A ( G ) of the graph G , let x = ( x 1 , . . . , x n ) ∈ R n + b e an nonnegativ e eigen vector corresp onding to ρ t ( G ) with x t 1 + · · · + x t n = 1. Then ρ t ( G ) = A ( G ) x t = t X { i 1 ,i 2 ,...,i t }∈ C t ( G ) x i 1 x i 2 · · · x i t . H˝ older’s inequality shows that for tw o nonnegative vectors x = ( x 1 , . . . , x n ) T and y = ( y 1 , . . . , y n ) T , if tw o p ositive n umber p and q satisfy 1 p + 1 q = 1, then P n i =1 x i y i ≤ ( P n i =1 x p i ) 1 p ( P n i =1 y q i ) 1 q , the equalit y holds if and only if x and y are prop ortional. Thus, w e hav e ρ t ( G ) = t X { i 1 ,i 2 ,...,i t } = I ∈ C t ( G )  α ( I ) t  ( α ( I )) t ! 1 t ( α ( I )) t  α ( I ) t  ! 1 t x i 1 x i 2 · · · x i t ≤ t   X I ∈ C t ( G )  α ( I ) t  ( α ( I )) t ! 1 t − 1   t − 1 t   X { i 1 ,i 2 ,...,i t } = I ∈ C t ( G ) ( α ( I )) t  α ( I ) t  x t i 1 x t i 2 · · · x t i t   1 t . Since x t 1 + x t 2 + · · · + x t n = 1, b y Lemma 2.4, w e ha v e ρ t ( G ) ≤ t   X I ∈ C t ( G )  α ( I ) t  ( α ( I )) t ! 1 t − 1   t − 1 t . Next, w e characterize all graphs attaining equality in the ab ov e inequalit y . Ac- 7 cording to the pro of ab o ve, by H˝ older’s inequality , equalit y holds if and only if X { i 1 ,i 2 ,...,i t } = I ∈ C t ( G ) ( α ( I )) t  α ( I ) t  x t i 1 x t i 2 · · · x t i t = 1 and for eac h t -clique I = { i 1 , i 2 , . . . , i t } ∈ C t ( G ), x i 1 x i 2 · · · x i t = c  α ( I ) t  ( α ( I )) t ! 2 t for some constant c > 0. By Lemma 2.4 and x i > 0 for an y v ertex i con tained in a t -clique of G , the equalit y is equiv alent to the follo wing: (1) The graph obtained from G b y deleting edges not con tained in t -cliques is a complete ω -partite graph (p ossibly with some isolated vertices), and its vertex classes V 1 , V 2 , . . . , V ω satisfy P v ∈ V i x t v = 1 ω for all i ∈ [ ω ]. (2) And for eac h t -clique { i 1 , i 2 , · · · , i t } ∈ C t ( G ), x i 1 x i 2 · · · x i t = c ′ for some con- stan t c ′ > 0. This completes the pro of for ω = t . Considering ω ≥ t + 1, and let G b e a complete ω - partite graph. F or any i  = j and u ∈ V i , v ∈ V j , b y item (2), we hav e x u x i 2 · · · x i t = c ′ = x v x i 2 · · · x i t for any i 2 , · · · , i t / ∈ ( V i ∪ V j ) and i 2 , · · · , i t resp ectiv ely come from other t − 1 differen t partitions. Then x u = x v , and therefore | V 1 | = | V 2 | = · · · = | V ω | , completing the pro of. Remark 3.2. When t = 2 , the c onclusion in The or em 3.1 shows that ρ 2 ( G ) ≤ 2 X e ∈ E ( G ) α ( e ) − 1 α ( e ) , which is the lo c alize d version of sp e ctr al T ur´ an ’s the or em (i.e., The or em 1.4). A nd α ( I ) ≤ ω ( G ) for any I ∈ C t ( G ) , then fr om The or em 3.1, we have  ρ t ( G ) t  t ≤ 1 ω ( G ) t  ω ( G ) t  | C t ( G ) | t − 1 , which implies the ine quality in The or em 1.5. As we all kno w, for a complete bipartite graph with partition V 1 and V 2 , its sp ectral radius is equal to p | V 1 | · | V 2 | . F rom Theorem 3.1, w e can get the t -clique 8 sp ectral radius of a complete t -partite graph directly . Corollary 3.3. [12] L et G b e a c omplete t -p artite gr aph with p artition V 1 , V 2 , . . . , V t . Then ρ t ( G ) = t Y i =1 | V i | ! t − 1 t . F or any t -clique I = { i 1 , i 2 , . . . , i t } ∈ C t ( G ), since α ( I ) ≤ α ( i j ) for j = 1 , . . . , t , w e can get the following conclusion. Corollary 3.4. L et G b e a gr aph with t -clique sp e ctr al r adius ρ t ( G ) and clique numb er ω and let 2 ≤ t ≤ ω . Then ( ρ t ( G )) t ≤ t   X v ∈ V ( G ) c t ( v ) t − 1 s  α ( v ) t  ( α ( v )) − t   t − 1 . Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c ontaine d in t -cliques is a c omplete t -p artite gr aph for ω = t , or a c omplete r e gular ω -p artite gr aph for ω ≥ t + 1 (p ossibly with some isolate d vertic es). Pr o of. Observ e that for any t -clique I = { i 1 , i 2 , . . . , i t } ∈ C t ( G ), α ( I ) ≤ α ( i j ) for j = 1 , . . . , t . Th us, b y Theorem 3.1, we hav e  ρ t ( G ) t  t ≤   X I ∈ C t ( G ) t − 1 s  α ( I ) t  ( α ( I )) t   t − 1 ≤ 1 t t − 1   X { i 1 ,i 2 ,...,i t } = I ∈ C t ( G ) t X j =1 t − 1 s  α ( i j ) t  ( α ( i j )) t   t − 1 = 1 t t − 1   X v ∈ V ( G ) c t ( v ) t − 1 s  α ( v ) t  ( α ( v )) t   t − 1 . Next,w e consider the equalit y . By Theorem 3.1, the equality holds in the last inequalit y if and only if the graph obtained from G by deleting edges not contained in t -cliques is a complete t - partite graph for ω = t , or a complete regular ω -partite graph for ω ≥ t + 1 (p ossibly with some isolated v ertices). In this condition, for any t -clique I = { i 1 , i 2 , . . . , i t } ∈ C t ( G ), α ( I ) = α ( i j ) = ω for j = 1 , . . . , t , the equalities also hold in the further inequality , completing the proof. 9 When t = 2, Corollary 3.4 is a lo calized v ersion of Wilf ’s equalit y given in [14]. The following result can provide a new upp er b ound on t -clique spectral radius based on α ( v ). Theorem 3.5. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . Then X v ∈ V ( G ) c t ( v ) t t − 1 s  α ( v ) t  ( α ( v )) − t ≤   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) − t   t . Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c on- taine d in t -cliques is a c omplete r e gular ω -p artite gr aph (p ossibly with some isolate d vertic es). Pr o of. A sp ecific form of Muirhead’s inequality states that for p ositive real num b er z 1 , z 1 , . . . , z m , m X k =1 1 z k ≤ m X k =1 z m − 1 k ! 1 Q m j =1 z j , and equality holds if and only if z 1 = · · · = z m . Then, w e ha ve X v ∈ V ( G ) c t ( v ) t t − 1 s  α ( v ) t  ( α ( v )) t = 1 t X { i 1 ,i 2 ,...,i t }∈ C t ( G )   t X j =1 t − 1 s  α ( i j ) t  ( α ( i j )) t   ≤ 1 t X { i 1 ,i 2 ,...,i t }∈ C t ( G )   t X j =1 ( α ( i j )) t  α ( i j ) t  ! t Y j =1 t − 1 s  α ( i j ) t  ( α ( i j )) t   = t X { i 1 ,i 2 ,...,i t }∈ C t ( G )   1 t 2 t X j =1 ( α ( i j )) t  α ( i j ) t  ! t Y j =1 t − 1 s  α ( i j ) t  ( α ( i j )) t   . Let V ( G ) = { v 1 , v 2 , . . . , v n } . W e construct a vector x =   t − 1 s  α ( v 1 ) t  ( α ( v 1 )) t , t − 1 s  α ( v 2 ) t  ( α ( v 2 )) t , . . . , t − 1 s  α ( v n ) t  ( α ( v n )) t   T ∈ R n + . And let W ( G ) = ( w i 1 i 2 ··· i t ) b e the w eigh t t -clique tensor of G with w i 1 i 2 ··· t t = ω i 1 i 2 ··· i t 1 ( t − 1)! if { i 1 , i 2 , . . . , i t } is a t -clique in G and w i 1 i 2 ··· t t = 0 otherwise, where 10 ω i 1 i 2 ··· i t = 1 t 2 P t j =1 ( α ( i j )) t ( α ( i j ) t ) . Then X v ∈ V ( G ) c t ( v ) t t − 1 s  α ( v ) t  ( α ( v )) t ≤ W ( G ) x t . Let y = x || x || 1 ∈ R n + . Then || y || 1 = P n i =1 y i = 1. By Lemma 2.5, w e ha ve W ( G ) x t = W ( G ) y t · || x || t 1 ≤ || x || t 1 . Th us, X v ∈ V ( G ) c t ( v ) t t − 1 s  α ( v ) t  ( α ( v )) t ≤ W ( G ) x t ≤ || x || t 1 =   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) t   t . Next, w e consider the equalit y in the ab o v e inequality . If the graph obtained from G b y deleting edges not con tained in t -cliques is a complete regular ω -partite graph, w e will v erify that equalit y holds. Without loss of generalit y , assume that ω | n and G is a complete regular ω -partite graph with. Then α ( v ) = ω and c t ( v ) =  ω − 1 t − 1   n ω  t − 1 for any v ∈ V ( G ). Therefore, we can pro of that the equality holds. Con versely , if equality holds, then by Lemma 2.5, G [supp( y )] is a complete ω - partite graph with partition V 1 , V 2 , . . . , V ω satisfy P v ∈ V i y v = 1 ω for all i ∈ [ ω ]. By Muirhead’s inequalit y , w e kno w that cl ( v ) = ω for eac h v contained in t -cliques. Hence, for an y i ∈ [ ω ], 1 ω = 1 || x || 1 X v ∈ V i t − 1 s  α ( v ) t  ( α ( v )) t = | V i | || x || 1 t − 1 s  ω t  ω t , i.e., | V 1 | = | V 2 | = · · · = | V ω | . Consequently , the graph obtained from G b y deleting edges not con tained in t -cliques is a complete regular ω -partite graph. By the ab ov e theorem, a new upp er bound for ρ t ( G ) is giv en as follo ws. When t = 2, this is the conclusion given in [14], which serv es as a lo cal version of Wilf ’s theorem. Corollary 3.6. L et G b e a gr aph with t -clique sp e ctr al r adius ρ t ( G ) and clique 11 numb er ω and let 2 ≤ t ≤ ω . Then ρ t ( G ) ≤ t   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) − t   t − 1 . Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c on- taine d in t -cliques is a c omplete r e gular ω -p artite gr aph (p ossibly with some isolate d vertic es). Pr o of. By Corollary 3.4, w e ha v e  ρ t ( G ) t  t ≤ 1 t t − 1   X v ∈ V ( G ) c t ( v ) t − 1 s  α ( v ) t  ( α ( v )) t   t − 1 =   X v ∈ V ( G ) c t ( v ) t t − 1 s  α ( v ) t  ( α ( v )) t   t − 1 ≤   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) t   t ( t − 1) . By Corollary 3.4 and Theorem 3.5, w e can c haracterize all graphs attaining equalities in the ab o v e inequalities. Com bining Lemma 2.2 and Corollary 3.4, a upp er bound on the num b er of t - cliques can b e giv en directly . Corollary 3.7. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . Then | C t ( G ) | ≤ n   1 t X v ∈ V ( G ) c t ( v ) t − 1 s  α ( v ) t  ( α ( v )) − t   t − 1 t Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c on- taine d in t -cliques is a c omplete r e gular ω -p artite gr aph (p ossibly with some isolate d vertic es). Com bining Theorem 3.5 and Corollary 3.7, we can get a weak er upp er b ound on the num b er of t -cliques but without c t ( v ). 12 Corollary 3.8. L et G b e an n -vertex gr aph with clique numb er ω and let 2 ≤ t ≤ ω . Then | C t ( G ) | ≤ n   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) − t   t − 1 . Equality holds if and only if the gr aph obtaine d fr om G by deleting e dges not c on- taine d in t -cliques is a c omplete r e gular ω -p artite gr aph (p ossibly with some isolate d vertic es). Remark 3.9. F r om Cor ol lary 3.8, by H˝ older’s ine quality, we have | C t ( G ) | ≤ n   X v ∈ V ( G ) t − 1 s  α ( v ) t  ( α ( v )) t   t − 1 ≤ n      X v ∈ V ( G ) 1 t − 1 t − 2   t − 2 t − 1   X v ∈ V ( G )  α ( v ) t  ( α ( v )) t   1 t − 1    t − 1 = n t − 1   X v ∈ V ( G )  α ( v ) t  ( α ( v )) t   , which is a vertex-b ase d lo c alize d Zykov’s ine quality given in [1]. By H˝ older’s ine qual- ity, e quality holds in the se c ond ine quality if and only if ( α ( v ) t ) ( α ( v )) t = c for e ach vertex v c ontaine d in t -cliques and some c onstant c > 0 , which is e quivalent to the or der of the lar gest clique c ontaining e ach vertex in G is e qual. Thus, if ther e exist two vertic es u, v ∈ V ( G ) with α ( u ) ≥ t , α ( v ) ≥ t and α ( u )  = α ( v ) , then the upp er b ound of the numb er t -cliques given in Cor ol lary 3.8 is strictly less than that in [1]. Ac knowledgemen ts References [1] R. Adak and L. Chandran. Generalized Zyko v’s Theorem. 2025. [2] N. Alon and P . F rankl. T ur´ an graphs with bounded matching num b er. J. Comb. The ory, Ser. B , 165:223–229, 2024. 13 [3] N. Alon and C. Shikhelman. Man y T copies in H -free graphs. J. Combin. The ory Ser. B , 121:146–172, 2016. [4] L. Arag˜ ao and V. Souza. Localised graph Maclaurin inequalities. A nn. Comb. , 28:1021–1033, 2024. [5] D. Bradaˇ c. A generalization of Tur´ an’s theorem. 2022. [6] S. Cioabˇ a, L. F eng, M. T ait, and X. Zhang. The maxim um spectral radius of graphs without friendship subgraphs. Eur. J. Comb. , 27(4):# P4.22, 2020. [7] P . Erd˝ os. On the num b er of complete subgraphs contained in certain graphs. Magy. T ud. A kad. Mat. Kut. Int´ ez. K˝ ozl. , 7:459–474, 1962. [8] X. F ang, X. Zhu, and Y. Chen. Generalized Tur´ an problem for a path and a clique. Eur. J. Comb. , 127:104137, 2025. [9] D. Gerbner, E. Gy˝ ori, A. Methuku, and M. Vizer. Generalized Tur´ an problems for even cycles. J. Comb. The ory, Ser. B , 145:169–213, 2020. [10] R. Kirsc h and J. Nir. A lo calized approac h to generalized Tur´ an problems. Ele ctr on. J. Combin. , 31(3):# P3.34, 2024. [11] L. Lim. Singular v alues and eigen v alues of tensors: a v ariational approach. In 1st IEEE International Workshop on Computational A dvanc es in Multi-Sensor A daptive Pr o c essing , pages 129–132. IEEE, 2005. [12] C. Liu and C. Bu. On a generalization of the spectral Man tel’s theorem. J. Comb. Optim. , 46:14, 2023. [13] C. Liu and C. Bu. A tensor’s sp ectral bound on the clique n umber. Discr ete Math. , 349:114694, 2026. [14] L. Liu and B. Ning. A new sp ectral Tur´ an theorem for w eighted graphs and consequences. 2025. [15] L. Liu and B. Ning. Lo cal prop erties of the sp ectral radius and Perron v ector in graphs. J. Combin. The ory Ser. B , 176:241–253, 2026. [16] R. Luo. The maxim um n umber of cliques in graphs without long cycles. J. Comb. The ory, Ser. B , 128:219–226, 2018. 14 [17] J. Ma and Y. Qiu. Some sharp results on the generalized Tur´ an n umbers. Eur. J. Comb. , 84:103026, 2020. [18] D. Malec and C. T ompkins. Lo calized versions of extremal problems. Eur. J. Combin. , 112:103715, 2023. [19] V. Nikiforov. Some inequalities for the largest eigen v alue of a graph. Combin. Pr ob ab. Comput. , 11:179–189, 2002. [20] L. Qi. Eigen v alues of a real supersymmetric tensor. J. Symb olic Comput. , 40(6):1302–1324, 2005. [21] L. Qi. Symmetric nonnegativ e tensors and copositive tensors. Line ar A lgebr a Appl. , 439:228–238, 2013. [22] P . T ur´ an. Researc h problems. A Magyar T udom´ anyos A kad´ emia Matematikai Kutat´ o Int´ ezet´ enek K¨ ozlem´ enyei , 6(3):417–423, 1961. [23] J. W ang, L. Kang, and Y Xue. On a conjecture of sp ectral extremal problems. J. Comb. The ory, Ser. B , 159:20–41, 2023. [24] H. Wilf. The eigen v alues of a graph and its c hromatic n umber. J. L ondon Math. So c. , 42:330–332, 1967. [25] Z. Y an, B. Y ang, and Y. P eng. A sp ectral generalized Alon-Frankl theorem. Discr ete Math. , 349:114785, 2026. [26] L. Y u and Y. Peng. A spectral version of the theorem of Zyk o v and Erd˝ os. Ele ctr on. J. Combin. , 32(4):# P4.15, 2025. [27] A. Zyk ov. On some prop erties of linear complexes. Matematicheskii sb ornik , 66(2):163–188, 1949. 15