E-cospectral hypergraphs and some hypergraphs determined by their spectra
aa r X i v : . [ m a t h . C O ] J un E-cospectral hypergraphs and some hypergraphsdetermined by their spectra
Changjiang Bu a , Jiang Zhou a,b , Yimin Wei c a College of Science, Harbin Engineering University, Harbin 150001, PR China b College of Computer Science and Technology, Harbin Engineering University, Harbin150001, PR China c School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary AppliedMathematics, Fudan University, Shanghai, 200433, PR China
Abstract
Two k -uniform hypergraphs are said to be cospectral (E-cospectral), if theiradjacency tensors have the same characteristic polynomial (E-characteristicpolynomial). A k -uniform hypergraph H is said to be determined by its spec-trum, if there is no other non-isomorphic k -uniform hypergraph cospectralwith H . In this note, we give a method for constructing E-cospectral hyper-graphs, which is similar with Godsil-McKay switching. Some hypergraphsare shown to be determined by their spectra. Keywords:
Cospectral hypergraph, E-cospectral hypergraphs, Adjacencytensor
AMS classification:
1. Introduction
For a positive integer n , let [ n ] = { , . . . , n } . An order k dimension n tensor A = ( a i ··· i k ) ∈ C n ×···× n is a multidimensional array with n k entries,where i j ∈ [ n ], j = 1 , . . . , k . A is called symmetric if a i i ··· i k = a i σ (1) i σ (2) ··· i σ ( k ) for any permutation σ on [ k ]. We sometimes write a i ··· i k as a i α , where α = i · · · i k . When k = 1, A is a column vector of dimension n . When k = 2, A is an n × n matrix. The unit tensor of order k > n is the tensor I n = ( δ i i ··· i k ) such that δ i i ··· i k = 1 if i = i = · · · = i k , and Email addresses: [email protected] (Changjiang Bu), [email protected] (Jiang Zhou), [email protected] (Yimin Wei)
Preprint submitted to Elsevier September 17, 2018 i i ··· i k = 0 otherwise. When k = 2, I n is the identity matrix I n . Recently,Shao [17] introduce the following product of tensors, which is a generalizationof the matrix multiplication. Definition 1.1. [17]
Let A and B be order m > and order k > , dimension n tensors, respectively. The product AB is the following tensor C of order ( m − k −
1) + 1 and dimension n with entries: c iα ...α m − = X i ,...,i m ∈ [ n ] a ii ...i m b i α · · · b i m α m − , where i ∈ [ n ] , α , . . . , α m − ∈ [ n ] k − . Let A be an order m > n tensor, and let x = ( x , . . . , x n ) ⊤ .From Definition 1.1, the product A x is a vector in C n whose i -th componentis (see Example 1.1 in [17])( A x ) i = X i ,...,i m ∈ [ n ] a ii ··· i m x i · · · x i m . In 2005, the concept of tensor eigenvalues was posed by Qi [13] and Lim[11]. A number λ ∈ C is called an eigenvalue of A , if there exists a nonzerovector x ∈ C n such that A x = λx [ m − , where x [ m − = ( x m − , . . . , x m − n ) ⊤ .The determinant of A , denoted by det( A ), is the resultant of the systemof polynomials f i ( x , . . . , x n ) = ( A x ) i ( i = 1 , . . . , n ). The characteristicpolynomial of A is defined as Φ A ( λ ) = det( λ I n − A ), where I n is the unittensor of order m and dimension n . It is known that eigenvalues of A areexactly roots of Φ A ( λ ) (see [17]).For an order m > n tensor A , a number λ ∈ C is called an E-eigenvalue of A , if there exists a nonzero vector x ∈ C n such that A x = λx and x ⊤ x = 1. In [14], the E-characteristic polynomial of A is defined as φ A ( λ ) = Res x (cid:16) A x − λ ( x ⊤ x ) m − x (cid:17) m is even , Res x,β A x − λβ m − xx ⊤ x − β ! m is odd , where ‘Res’ is the resultant of the system of polynomials. It is known that E-eigenvalues of A are roots of φ A ( λ ) (see [14]). If m = 2, then φ A ( λ ) = Φ A ( λ )is just the characteristic polynomial of the square matrix A .2 hypergraph H is called k - uniform if each edge of H contains exactly k distinct vertices. All hypergraphs in this note are uniform and simple. Let K kn denote the complete k -uniform hypergraph with n vertices, i.e., every k distinct vertices of K kn forms an edge. For a k -uniform hypergraph H =( V ( H ) , E ( H )), a hypergraph G = ( V ( G ) , E ( G )) is a sub-hypergraph of H ,if V ( G ) ⊆ V ( H ) and E ( G ) ⊆ E ( H ). For any edge u · · · u k ∈ E ( H ), wesay that u k is a neighbor of { u , . . . , u k − } . The complement of H is a k -uniform hypergraph with vertex set V ( H ) and edge set E ( K k | V ( H ) | ) \ E ( H ).The adjacency tensor of H , denoted by A H , is an order k dimension | V ( H ) | tensor with entries (see [2]) a i i ··· i k = ( k − if i i · · · i k ∈ E ( H ) , . Clearly A H is a symmetric tensor. We say that two k -uniform hypergraphsare cospectral ( E-cospectral ), if their adjacency tensors have the same charac-teristic polynomial (E-characteristic polynomial). A k -uniform hypergraph H is said to be determined by its spectrum , if there is no other non-isomorphic k -uniform hypergraph cospectral with H . We shall use “DS” as an abbrevia-tion for “determined by its spectrum” in this note. Cospectral (E-cospectral)hypergraphs and DS hypergraphs are generalizations of cospectral graphs andDS graphs in the classic sense [3].Recently, the research on spectral theory of hypergraphs has attractedextensive attention [2,6-9,12,15-20]. In this note, we give a method for con-structing E-cospectral hypergraphs. Some hypergraphs are shown to be DS.
2. Preliminaries
The following lemma can be obtained from equation (2.1) in [17].
Lemma 2.1.
Let A = ( a i ··· i m ) be an order m > dimension n tensor, andlet P = ( p ij ) be an n × n matrix. Then ( P A P ⊤ ) i ··· i m = X j ,...,j m ∈ [ n ] a j ··· j m p i j p i j · · · p i m j m . We can obtain the following lemma from Lemma 2.1.
Lemma 2.2.
Let B = P A P ⊤ , where A is a tensor of dimension n , P is an n × n matrix. If A is symmetric, then B is symmetric. B = P A P ⊤ , where A is a tensor of dimension n , P is an n × n real orthogonal matrix. In [17], Shao pointed out that A , B are orthogonallysimilar tensors defined by Qi [13]. Orthogonally similar tensors have thefollowing property. Lemma 2.3. [10]
Let B = P A P ⊤ , where A is a tensor of dimension n , P is an n × n real orthogonal matrix. Then A and B have the same E-characteristic polynomial. A simplex in a k -uniform hypergraph is a set of k + 1 vertices where everyset of k vertices forms an edge (see [2, Definition 3.4]). Lemma 2.4.
Let G and H be cospectral k -uniform hypergraphs. Then G and H have the same number of vertices, edges and simplices.Proof. The degree of the characteristic polynomial of an order k dimension n tensors is n ( k − n − (see [13]). Since A G and A H are order k tensors, G and H have the same number of vertices. From [2, Theorem 3.15] and [2,Theorem 3.17], we know that G and H have the same number of edges andsimplices.
3. Main results
Let H = ( V ( H ) , E ( H )) be a k -uniform hypergraph with a partition V ( H ) = V ∪ V , and H satisfies the following conditions:(a) For each edge e ∈ E ( H ), e contains at most one vertex in V .(b) For any k − u , . . . , u k − ∈ V , { u , . . . , u k − } haseither 0 , | V | or | V | neighbors in V .Similar with GM switching [4, 5], we construct a hypergraph E-cospectralwith H as follows. Theorem 3.1.
Let H be a k -uniform hypergraph satisfies the conditions (a)and (b) described above. For any { u , . . . , u k − } ⊆ V which has | V | neigh-bors in V , by replacing these | V | neighbors with the other | V | vertices in V , we obtain a k -uniform hypergraph G which is E-cospectral with H .Proof. Let P = (cid:18) n J − I n I n (cid:19) , where n = | V | , n = | V | , J is the n × n all-ones matrix, n J − I n and I n correspond to the vertex sets V and V ,respectively. Then P = P ⊤ = P − . Suppose that A H = ( a i i ··· i k ), and4et B = P A H P ⊤ . By Lemma 2.2, B is symmetric. We need to show that B = A G . By Lemma 2.1, we have( B ) i ··· i k = X j ,...,j k ∈ V ( H ) a j ··· j k p i j p i j · · · p i k j k . (1)Note that P = (cid:18) n J − I n I n (cid:19) . From Eq. (1), we have( B ) i ··· i k = a i ··· i k if i , . . . , i k ∈ V . (2)Since H satisfies the condition (a), we have a j ··· j k = 0 if |{ j , . . . , j k }∩ V | > B ) i ··· i k = a i ··· i k = 0 if |{ i , . . . , i k } ∩ V | > . (3)Next we consider the case |{ i , . . . , i k } ∩ V | = 1. Note that B is symmetric.Without loss of generality, suppose that i ∈ V , i , . . . , i k ∈ V . From Eq.(1), we have( B ) i ··· i k = X j ∈ V a j i ··· i k p i j ( i ∈ V , i , . . . , i k ∈ V ) . (4)Since H satisfies the condition (b), S i ··· i k = { a j i ··· i k | j ∈ V , a j i ··· i k = 0 } contains either 0 , | V | or | V | elements for any given i , . . . , i k ∈ V . Bycomputing the sum in (4), we have( B ) i ··· i k = a i ··· i k = 0 if i ∈ V , i , . . . , i k ∈ V , | S i ··· i k | = 0 . (5)( B ) i ··· i k = a i ··· i k = 1( k − i ∈ V , i , . . . , i k ∈ V , | S i ··· i k | = | V | . (6)( B ) i ··· i k = 0 if i ∈ V , i , . . . , i k ∈ V , a i ··· i k = 1( k − , | S i ··· i k | = 12 | V | . (7)( B ) i ··· i k = 1( k − i ∈ V , i , . . . , i k ∈ V , a i ··· i k = 0 , | S i ··· i k | = 12 | V | . (8)From Eqs. (2)(3) and (5)-(8), we have B = P A H P ⊤ = A G . By Lemma 2.3, G is E-cospectral with H . 5f two k -uniform hypergraphs G and H are isomorphic, then there existsa permutation matrix P such that A G = P A H P ⊤ (see [1, 17]). From Lemma2.3, we know that two isomorphic k -uniform hypergraphs are E-cospectral.By using the method in Theorem 3.1, we give a class of non-isomorphicE-cospectral hypergraphs as follows. Example.
Let H be a 3-uniform hypergraph whose vertex set and edge setare V ( H ) = { u , u , u , u , v , . . . , v n } ( n > ,E ( H ) = { v v u , v v u , v v u , v v u , v v u , v v u } ∪ F, where each edge in F contains three vertices in { v , . . . , v n } , and each vertexin { v , . . . , v n } is contained in at least one edge in F if n >
4. Let G be a3-uniform hypergraph whose vertex set and edge set are V ( G ) = V ( H ) , E ( G ) = { v v u , v v u , v v u , v v u , v v u , v v u } ∪ F. The vertex set V ( H ) has a partition V ( H ) = V ∪ V such that H and G satisfythe conditions in Theorem 3.1, where V = { u , u , u , u } , V = { v , . . . , v n } .Then G and H are E-cospectral. Moreover, G and H are non-isomorphic E-cospectral hypergraphs, because H has an isolated vertex u and G has noisolated vertices.Let K kn − e denote the k -uniform hypergraph obtained from K kn by deletingone edge. We can obtain the following result from Lemma 2.4. Theorem 3.2.
The complete k -uniform hypergraph K kn , the hypergraph K kn − e and their complements are DS. Any k -uniform sub-hypergraph of K kk +1 isDS. The disjoint union of K kk +1 and some isolated vertices is DS. If G is a sub-hypergraph of a hypergraph H , then let H \ G denote thehypergraph obtained from H by deleting all edges of G . Theorem 3.3.
The hypergraph K kn \ G is DS, where G is a k -uniform sub-hypergraph of K kn such that all edges of G share k − common vertices.Proof. Let H be any k -uniform hypergraph cospectral with K kn \ G . Supposethat G has r edges. By Lemma 2.4, H can be obtained from K kn by deleting r edges. Deleting r edges from K kn destroys at least P r − i =0 ( n − k − i ) simplices,with equality if and only if all deleted edges share k − H = K kn \ G . 6 cknowledgements. This work is supported by the National Natural Science Foundation ofChina (No. 11371109 and No. 11271084), and the Fundamental ResearchFunds for the Central Universities.
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