The bounds of the spectral radius of general hypergraphs in terms of clique number
aa r X i v : . [ m a t h . C O ] J u l The bounds of the spectral radius of generalhypergraphs in terms of clique number ∗ Cunxiang Duan a,b , Ligong Wang a,b, † a School of Mathematics and Statistics, Northwestern Polytechnical University,Xi’an, Shaanxi 710129, P.R. China. b Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University,Xi’an, Shaanxi 710129, P.R. China.E-mail: [email protected]; [email protected]
Abstract
The spectral radius (or the signless Laplacian spectral radius) of a gen-eral hypergraph is the maximum modulus of the eigenvalues of its adja-cency (or its signless Laplacian) tensor. In this paper, we firstly obtaina lower bound of the spectral radius (or the signless Laplacian spectralradius) of general hypergraphs in terms of clique number. Moreover,we present a relation between a homogeneous polynomial and the cliquenumber of general hypergraphs. As an application, we finally obtain anupper bound of the spectral radius of general hypergraphs in terms ofclique number.
Key Words : spectral radius, clique number, general hypergraphs
AMS Subject Classification (2020) : 15A42, 05C50.
Let G be a general hypergraph with vertex set V = V ( G ) = { v , v , . . . , v n } and edgeset E = E ( G ) = { e , e , . . . , e m } , where e , e , . . . , e m ⊆ V ( G ) . If V ′ ⊆ V and E ′ ⊆ E ,then H = ( V ′ , E ′ ) is called a subhypergraph of G. The rank (resp., co-rank) of G is rank ( G ) = max {| e | : e ∈ E } (resp., corank ( G ) = min {| e | : e ∈ E } ). If the rank and the ∗ Supported by the National Natural Science Foundation of China (No. 11871398) and the Seed Foun-dation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (No.CX2020190). † Corresponding author. G are equal to k , then G is called k -uniform. Obviously, a graphis a 2-uniform hypergraph. For two vertices u, v ∈ V, if there exists an edge e such that u, v ∈ e ∈ E, then vertices u and v are said to be adjacent. Otherwise, two vertices u and v are said to be nonadjacent. We use E v to denote the set of edges containing v of G. Thedegree of a vertex v in G is d v = | E v | . In addition, we use R ( v ) to denote the multiset ofedge types in E v for a vertex v. Let G = ( V, E ) be a general hypergraph. The set R = {| e | : e ∈ E } is called the set ofedge types of G. Then G is also called an R -graph. For a set S and a positive integer i , let (cid:0) Si (cid:1) = { T ⊆ S : | T | = i } . A complete R -graph on n vertices is an R -graph with vertex set[ n ] and edge set ∪ i ∈ R (cid:0) [ n ] i (cid:1) , where [ n ] = { , , . . . , n } . A clique of a R -graph G is a complete R -subgraph of G . A maximal clique is a clique that cannot extended to a larger clique,and a maximal clique is a clique which covers as many vertices as possible. The cliquenumber ω ( G ) of a hypergraph G is the number of vertices in a maximum clique of G . If E ( G ) = ∅ , then we define ω ( G ) = 1 . In 2005, the concept of tensor eigenvalues and the spectra of tensors were independentlyintroduced by Qi [18] and Lim [13]. A tensor A = ( a i i ...i k ) with order k and dimension n isa multidimensional array, where a i i ...i k ∈ C and 1 ≤ i , i , . . . , i k ≤ n. For an n -dimensioncomplex vector x = ( x , x , . . . , x n ) T , A x k − is defined as an n -dimension complex vectorwhose i -th component is the following( A x k − ) i = n X i ,...,i k =1 a ii ...i k x i · · · x i k , f or any i ∈ [ n ] . Let x [ k − = ( x k − , x k − , . . . , x k − n ) T ∈ C n . Then a number λ ∈ C is called an eigenvalueof the tensor A if there exists a nonzero vector x ∈ C n such that A x k − = λx [ k − , and in this case, x is called an eigenvector of A corresponding to the eigenvalue λ. Thespectral radius of A is ρ ( A ) = max {| λ | : λ is an eigenvalue of A} . In particular, if x isreal, then λ is called an H -eigenvalue. If x ∈ R n + , then λ is called an H + -eigenvalue. If x ∈ R n ++ , then λ is called an H ++ -eigenvalue [19]. Definition 1.1. ([1]) Let G = ( V, E ) be a general hypergraph with n vertices, m edges and rank ( G ) = k. The adjacency tensor A ( G ) of G is A ( G ) = ( a i ,i ,...,i k ) , ≤ i , i , . . . , i k ≤ n. For all edges e = { v l , v l , . . . , v l s } ∈ E of cardinality s ≤ k,a i ,i ,...,i k = sα ( s ) , where α ( s ) = X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! , and i , i , . . . , i k are chosen in all possible ways from L = { l , . . . , l s } with at least once foreach element of the set L, while each k i represents the times that l i appears in { i , i , . . . , i k } .The other positions of the tensor are zero. G refer to the eigenvalues of the adjacencytensor of G. And we use ρ ( G ) to denote the spectral radius of a general hypergraph G. Let G be a general hypergraph with n vertices and rank ( G ) = k , and x = ( x , x , . . . , x n ) T be a vector. For an edge e = { i , i , . . . , i s } ∈ E, we have x T ( A ( e ) x k − ) = sα ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s . And we also have x T ( A ( G ) x k − ) = X e ∈ E ( G ) x T ( A ( e ) x k − ) . Let D ( G ) be a degree diagonal tensor with order k and dimensional n , and its diagonalelement d ii...i (or simply d i ) be the degree of a vertex v i of G , for all i ∈ { , , . . . , n } . Then Q ( G ) = D ( G ) + A ( G ) is the signless Laplacian tensor of the hypergraph G . The signlessLaplacian eigenvalues of a hypergraph G refer to the eigenvalues of the signless Laplaciantensor of G . We use q ( G ) to denote the signless Laplacian spectral radius of G .Similarly, Let G be a hypergraph with n vertices and rank ( G ) = k , and x = ( x , x , . . . , x n ) T be a vector. For an edge e = { i , i , . . . , i s } ∈ E, we have x T ( Q ( e ) x k − ) = s X j =1 x ki j + sα ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s . And we also have x T ( Q ( G ) x k − ) = X e ∈ E x T ( Q ( e ) x k − ) . In 1965, Motzkin and Straus [12] defined a homogeneous polynomial L ( G, x ) of a graph G. Definition 1.2. ([12]) Let G = ( V, E ) be a graph with n vertices, and let x = ( x , x , . . . , x n ) be a nonnegative real vector such that P ni =1 x i = 1 . Then L ( G, x ) = X { i,j }∈ E x i x j . Moreover, Motzkin and Straus [12] established a remarkable connection between theclique number and a homogeneous polynomial L ( G ) = max x ∈ S L ( G, x ) of a graph G, where S = (cid:8) x : x ∈ R n + , P ni =1 x i = 1 (cid:9) . In 2020, Hou et al. [11] first defined a homogeneous polynomial L ( G, x ) for a gen-eral hypergraph G , which generalized the definition L ( G, x ) from a graph G to a generalhypergraph G. efinition 1.3. ([11]) Let G = ( V, E ) be a general hypergraph with n vertices and rank ( G ) = k, and let x = ( x , x , . . . , x n ) be a nonnegative real vector such that P ni =1 x i = 1 . Then L ( G, x ) = X e ∈ E α ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s , where s is the number of vertices of e and α ( s ) = P k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! ··· k s ! . Similarly, for a general hypergraph
G,L ( G ) = max x ∈ S L ( G, x ) , where S = (cid:8) x : x ∈ R n + , P ni =1 x i = 1 (cid:9) . In 1986, Wilf [20] gave a lower bound on the spectral radius of a graph in terms of cliquenumber, which was inspired by a result of Motzkin and Straus [12]. In 2007, Bollob´as andNikiforov [2] obtained a number of relations between the number of cliques of a graph G and the spectral radius. In 2007, Lu et al. [16] presented some lower and upper boundsfor the independence number and the clique number involving the Laplacian eigenvaluesof a graph G. In 2008, Liu and Liu [15] obtained some lower and upper bounds for theindependence number and the clique number involving the signless Laplacian eigenvaluesof a graph G. In 2009, Rota and Pelillo [3] gave some new upper and lower bounds on theclique number of graphs. In 2009, Nikiforov [17] gave some new bounds for the clique andindependence numbers of a graph in terms of its eigenvalues. In 2013, He et al. [10] gavethe bounds on the signless Laplacian spectral radius of graphs in terms of clique number,which disprove the two conjectures on the signless Laplacian spectral radius in [9].Recently, spectral hypergraph theory develops rapidly. There are many work about thespectral theory of hypergraphs [4, 5, 6, 14, 21, 22]. However, there are still few studiesthe relation bewteen the clique number and the (signless Laplacian) spectral radius of hy-pergraphs. In 2015, Xie and Qi [23] mainly researched some inequality relations betweenthe signless (Laplacian) H -eigenvalues and the clique (coclique) numbers of uniform hy-pergraphs. In 2020, Hou et al. [11] first defined a homogeneous polynomial for a generalhypergraph, and gave a Motzkin-Straus type result for { k − , k } -graphs. And they gavesome lower and upper bounds on the spectral radius of { k − , k } -graphs in terms of cliquenumber. In this paper, motivated by [11] and [23], we mainly generalize the results of Houet al. [11] form { k − , k } -graphs to R -graphs.This paper is organized as follows. In Section 2, some necessary lemmas and inequalitiesare given. In Section 3, we give a bound on the signless Laplacian spectral radius of generalhypergraphs in terms of clique number. We also present a relation between a homogeneouspolynomial and the clique number of general hypergraphs. Moreover, we also obtain alower and a upper bound on the spectral radius of general hypergraphs in terms of cliquenumber, respectively. 4 Preliminaries
In this section, we mainly give some useful lemmas and inequalities.
Lemma 2.1. (Maclaurin’s inequality [8]) Let x , x , . . . , x n be positive real numbers, and S k = P ≤ i
The following Theorem 3.1 presents a lower bound on the signless Laplacian spectral radiusof a general hypergraph in terms of clique number. This result generalizes Theorem 3.1 ofXie and Qi [23] from uniform hypergraphs to general hypergraphs.
Theorem 3.1.
Let G be an R -graph with clique number ω. Then q ( G ) ≥ X s ∈ R (cid:18) ω − s − (cid:19) , and the equality holds if and only if G is a complete R -graph. Proof.
Assume that G ′ is the maximum complete R -subgraph of G with ω vertices and rank ( G ) = k. Let x be a vector such that x i = k √ ω for i ∈ V ( G ′ ) and x i = 0 for otherwise.It is obvious that k x k kk = 1 . By the definition of the signless Laplacian spectral radius ofgeneral hypergraphs, we have q ( G ) ≥ x T ( Q ( G ) x k − )= X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R ( x ki + x ki + · · · + x ki s + sα ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s x k i x k i · · · x k s i s )= X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R ( 1 ω + 1 ω + · · · + 1 ω + 1 ω sα ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! )= X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R ( sω + 1 ω sα ( s ) α ( s ))= X s ∈ R X e ∈ E ( G ′ ) | e | = s sω = X s ∈ R sω (cid:18) ωs (cid:19) = 2 X s ∈ R (cid:18) ω − s − (cid:19) . (1)If q ( G ) = 2 P s ∈ R (cid:0) ω − s − (cid:1) , by Inequality (1), then we have q ( G ) = x T Q ( G ) x. Since Lemma2.2 holds for q ( G ) of a general hypergraph G , we know x is a positive vector. Therefore,we have ω = n. That is, G is a complete R -graph.If G is a complete R -graph, then we have ω = n. By the definition of the signlessLaplacian spectral radius of general hypergraphs, we have q ( G ) = 2 P s ∈ R (cid:0) ω − s − (cid:1) . (cid:3) If G be a k -uniform hypergraph with clique number ω , by Lemma 2.3, then we have ρ ( G ) ≥ (cid:0) ω − k − (cid:1) . Therefore, we only consider R -graphs such that | R | > heorem 3.2. Let G be an R -graph with clique number ω , rank ( G ) = k and corank ( G ) = c. Then ρ ( G ) ≥ ( ω − c + 1) c − ( ω − k + 1) k − c ( k − ω − c + 1) c − ( c − . Proof.
By Lemma 2.3, we have ρ ( G ) ≥ X s ∈ R (cid:18) ω − s − (cid:19) ≥ (cid:18) ω − k − (cid:19) + (cid:18) ω − c − (cid:19) = ( ω − k − ω − k )! + (cid:18) ω − c − (cid:19) = ( ω − c − ω − c )! ( c − ω − c )!( k − ω − k )! + (cid:18) ω − c − (cid:19) = (cid:18) ω − c − (cid:19) ( c − ω − c )!( k − ω − k )! + (cid:18) ω − c − (cid:19) ≥ ( ω − c + 1) c − ( c − c − ω − c )!( k − ω − k )! + ( ω − c + 1) c − ( c − ω − c + 1) c − ( k − ω − c ) · · · ( ω − k + 1) + ( ω − c + 1) c − ( c − ≥ ( ω − c + 1) c − ( ω − k + 1) k − c ( k − ω − c + 1) c − ( c − , where (cid:18) ω − c − (cid:19) = ( ω − c − ω − c )! = ( ω − · · · ( ω − c + 1)( c − ≥ ( ω − c + 1) c − ( c − . (cid:3) Remark. If c = k − >
1, by Theorem 3.2, then we have ρ ( G ) ≥ ( ω − k + 2) k − ( ω − k + 1)( k − ω − k + 2) k − ( k − ω − k + 2) k − ( k − (cid:18) ω − k + 1 k − (cid:19) = ω ( ω − k + 2) k − ( k − > ( ω − k + 2) k − ( k − . Thus, we know Theorem 3.2 improves Theorem 3.2 of Hou et al. [11].The following Theorem 3.3 generalizes Theorem 3.3 of Hou et al. [11] from { k − , k } -graphs to R -graphs. 7 heorem 3.3. Let G = ( V, E ) be an R -graph with clique number ω and rank ( G ) = k. If G is a complete R -graph, or if there exists two nonadjacent vertices v and v ′ in G suchthat R ( v ) = R ( v ′ ) , then L ( G ) = X s ∈ R ω k (cid:18) ωs (cid:19) . Proof.
Assume that G ′ is the maximum complete R -subgraph of G with ω vertices. Let x be an n -dimensional vector such that x i = ω for i ∈ V ( G ′ ) and x i = 0 for otherwise. Itis obvious that k x k = 1 . By the definition of L ( G ) , we have L ( G ) ≥ L ( G, x )= X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R α ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s = X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R ω k α ( s ) X k + ··· + k s = k,k ,...,k s ≥ k ! k ! k ! · · · k s != X e = { i ,i ,...,i s }∈ E ( G ′ ) s ∈ R ω k α ( s ) α ( s )= X s ∈ R X e ∈ E ( G ′ ) | e | = s ω k = X s ∈ R ω k (cid:18) ωs (cid:19) . In the following, we prove L ( G ) ≤ X s ∈ R ω k (cid:18) ωs (cid:19) (2)by induction on n .If n ≤ c − , then the edge set of G is empty. It is obvious that L ( G ) ≤ P s ∈ R ω k (cid:0) ωs (cid:1) holds. Assume that Inequality (2) holds for less than n vertices. Without loss of generality,supposed that n is sufficient large and x is a vector such that L ( G ) = L ( G, x ) . And x ≥ x ≥ · · · ≥ x t > x t +1 > · · · > x n = 0 . If t < n, then we have x n = 0 . We can obtain G from G by deleting the vertex n andthe edges containing the vertex n . Let G be a general hypergraph with clique number ω and ω ≤ ω. By the induction hypothesis, we have L ( G ) = L ( G ) ≤ X s ∈ R ω k (cid:18) ω s (cid:19) ≤ X s ∈ R ω k (cid:18) ωs (cid:19) , where (cid:16) ω (cid:17) k (cid:0) ω s (cid:1) is an monotonically increasing function of ω . t = n, then we have x ≥ x ≥ · · · ≥ x n > Case 1. If G is a complete R -graph, then we have ω = n. By the definition of L ( G )and triangle inequality, we have L ( G ) = max x ∈ S L ( G, x )= max x ∈ S X e = { i ,i ,...,is }∈ E ( G ) s ∈ R α ( s ) X k k ··· + ks = k,k ,...,ks ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s ≤ max x ∈ S X e ∈ E ( G ) | e | = c α ( c ) X k k ··· + kc = k,k ,...,kc ≥ k ! k ! k ! · · · k c ! x k i x k i · · · x k c i c + · · · + max x ∈ S X e ∈ E ( G ) | e | = k x i x i · · · x i k . (3)Without loss of generality, assume that there exist two integers d ≥ u ≥ k = sd + u for any s ∈ R. We know that α ( s ) = X k ,...,ks ≥ ,k ··· + ks = k k ! k ! k ! · · · k s != a (cid:18) s (cid:19) + a (cid:18) s (cid:19) + a (cid:18) s (cid:19) + · · · + a b (cid:18) su (cid:19) = a s !( s − a s !( s − a s !( s − · · · + a b s !( s − u )! u != s ! (cid:18) a ( s − a ( s − a ( s − · · · + a b ( s − u )! u ! (cid:19) , (4)where a = k !( k − s + 1)!1! · · · , a = ( k !( k − s )!2!1! ··· , if k − s = 2 , k !( k − s )!2!1! ··· , if k − s = 2 , , and a = ( k !( k − s − ··· , if k − s − , k !( k − s − ··· , if k − s − = 3 , , a b = k !( d + 1)! · · · ( d + 1)! | {z } u times d ! · · · d ! . Since G is a complete R -graph, we also have X e = { i ,i ,...,is }∈ E ( G ) , | e | = s X k k ··· + ks = k,k ,...,ks ≥ k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s k !( k − s + 1)!1! · · · X e = { i ,i ,...,i s }∈ E ( G ) x i x i · · · x i s (cid:0) x k − si + · · · + x k − si s (cid:1) + k !( k − s )!2!1! · · · X e = { i ,i ,...,i s }∈ E ( G ) x i x i · · · x i s (cid:0) x k − s − i x i + · · · + x i s − x k − s − i s (cid:1) + · · · + k !( d + 1)! · · · ( d + 1)! d ! · · · d ! X e = { i ,i ,...,i s }∈ E ( G ) x di x di · · · x di s (cid:0) x i · · · x i u + · · · + x i s − u +1 · · · x i s (cid:1) = k !( k − s + 1)! [ x k − s +11 X e = { ,i ,...,is }∈ E ( G ) , ≤ i < ··· If there exist two nonadjacent vertices v and v ′ in G such that R ( v ) = R ( v ′ ) , then we prove that there exists a vector y such that L ( G, y ) ≥ L ( G, x ) . By the definitionof L ( G ) , we denote L v ( G ) as L v ( G ) = X e ∈ E v , | e | = s ∈ R α ( s ) k ! k ! k ! · · · k s ! x k i x k i · · · x k s i s , for any a vertex v ∈ V ( G ) . Let y be a vector such that y i = x i for i = v, v ′ , y v = x v + x v ′ and y v ′ = 0 . And let z be a vector such that z i = x i for i = v and z v = x v ′ . It is obvious that y, z ∈ R n + . For anypositive integer k, we have ( x v + x v ′ ) k ≥ x kv + x kv ′ . Hence, we have L v ( G, y ) ≥ L v ( G, x ) + L v ( G, z ) . Moreover, without loss of generality, assume that L v ( G, z ) ≥ L v ′ ( G, x ) . Then we have L ( G, y ) − L ( G, x ) = L v ( G, y ) − L v ( G, x ) − L v ′ ( G, x ) ≥ L v ( G, z ) − L v ′ ( G, x ) ≥ . Thus, we can obtain G from G by deleting the vertex v ′ and the edges containing v ′ . Similarly, by the induction hypothesis, we have L ( G ) = L ( G, x ) ≤ L ( G, y ) ≤ L ( G ) ≤ X s ∈ R ( 1 ω ) k (cid:18) ωs (cid:19) . Therefore, we have L ( G ) = X s ∈ R ω k (cid:18) ωs (cid:19) , G is a complete R -graph, or there exists two nonadjacent vertices v and v ′ in G suchthat R ( v ) = R ( v ′ ) . (cid:3) The following Theorem 3.4 generalizes Theorem 3.4 of Hou et al. [11] from { k − , k } -graphs to R -graphs. Theorem 3.4. Let G = ( V, E ) be an R -graph with clique number ω and rank ( G ) = k. If G is a complete R -graph, or if there exists two nonadjacent vertices v and v ′ in G suchthat R ( v ) = R ( v ′ ) , then ρ ( G ) ≤ X s ∈ R k ( Uω ) k (cid:18) ωs (cid:19) , where U is the sum of entries of the principal eigenvector. Proof.