Jia-Yu Shao

Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs

A -uniform hypergraph is called odd-bipartite, if is even and there exists some proper subset of such that each edge of contains odd number of vertices in . Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected -uniform hypergraph are equal if and only if is even and is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected -uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal; thus, provide an answer to a question raised by L. Qi. By showing that the Cartesian product of two odd-bipartite -uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of is the sum of the Laplacian spectral radii of and , when and are both connected odd-bipartite.

Bounds on the kth eigenvalues of trees and forests

Abstract We prove the upper bound λk(T) ⩽ √[n / k] – 1 for the kth largest eigenvalue of a tree T (1 ⩽ k ⩽ [n / 2]), and show that this bound is best possible when n ≢ 0 (mod k). We further obtain the strict inequality √t-1-√ k -1 t -1 >λk(T)>√t-1 (where t= n k ) for the case n ≡ 0 (mod k) (2 ⩽ k ⩽ [n / 2]). Our upper bound is also proved to be the best possible upper bound for the kth largest eigenvalues of the forests in all cases.

Matrices with signed generalized inverses

Abstract A real matrix A is said to have a signed generalized inverse , if the sign pattern of its generalized inverse A + is uniquely determined by the sign pattern of A . In this paper, we give complete characterizations of the m×n matrices A which have signed generalized inverses, for both cases ρ(A)=n and ρ(A) (where ρ(A) is the term rank of A , and without loss of generality we assume n⩽m ), and thus solve a problem proposed in [B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056]. Using these characterizations, we are also able to show that the property of having a signed generalized inverse for a matrix A is inherited by all the submatrices B of A with ρ(B)=ρ(A) and is also inherited by all those matrices A 1 with ρ(A 1 )=ρ(A) which can be obtained from A by replacing some nonzero entries of A by zero. We also consider several special cases of a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, MA, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterization of the matrices in a special triangular block form to have signed generalized inverses.

On the Laplacian spectral radii of bicyclic graphs

A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n+1. Let B(n) be the set of all bicyclic graphs on n vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class B(n) (n>=7) together with the corresponding graphs.

Some new trace formulas of tensors with applications in spectral hypergraph theory

We give some graph theoretical formulas for the trace of a tensor which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the -uniform hypergraphs whose spectra are -symmetric, thus give an answer to a question raised in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292]. We generalize the results in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292, Theorem 4.2] and Hu and Qi [Discrete Appl. Math. 2014;169:140–151, Proposition 3.1] about the -symmetry of the spectrum of a -uniform hypergraph, and answer a question in Hu and Qi [Discrete Appl. Math. 2014;169:140–151] about the relation between the Laplacian and signless Laplacian spectra of a -uniform hypergraph when is odd. We also give a simplified proof of an expression for and discuss the expression for .

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131-135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.

The solution of a problem on matrices having signed generalized inverses

Abstract A real matrix A is said to have a signed generalized inverse , if the sign pattern of its generalized inverse A + is uniquely determined by the sign pattern of A . In this paper, we solve a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterizations of the matrices with a special lower triangular blocked form to have a signed generalized inverse. Using this characterization and the fact that every matrix which has a signed generalized inverse and full column term rank and no zero rows is permutation equivalent to a matrix of this form, we give two algorithms for determining whether or not a matrix with full column term rank, or a general matrix, has a signed generalized inverse.

Generalized exponents of primitive symmetric digraphs

Abstract A strongly connected digraph D of order n is primitive (aperiodic) provided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t , called the exponent of D , such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t . The exponent of D is the largest of n ‘generalized exponents’ that may be associated with D . If D is a symmetric digraph, then D is primitive if and only if its underlying graph is connected and is not bipartite. In this paper we determine the largest value of these generalized exponents over the set of primitive symmetric digraphs whose shortest odd cycle length is a fixed number r . We also characterize the extremal digraphs. Our results are common generalizations of a number of related results in the literature.

Matrices with Special Sign Patterns of Signed Generalized Inverses

A real matrix A is said to have a signed generalized inverse (GI) if the sign pattern of its GI A+ is uniquely determined by the sign pattern of A. We characterize those sign pattern matrices with a signed GI, and the GI of it is nonnegative, or is positive, or has no zeros.

On digraphs and forbidden configurations of strong sign nonsingular matrices

A square real matrix A called a strong sign nonsingular matrix (or “S2NS” matrix) if all matrices with the same sign pattern as A nonsingular and the inverses of these matrices all have the same sign pattern. A digraph which is the underlying digraph of the signed digraph of an S2NS matrix (with a negative main diagonal) is called an S2NS digraph. In [9], Thomassen gave a characterization of strongly connected S2NS digraphs in terms of the forbidden subdigraphs. In [2], Brualdi and Shader constructed minimal forbidden configurations for S2NS digraphs for the general cases where the digraphs considered are not necessarily strongly connected. They also proposed the problem about the existence of new minimal forbidden configurations other than those found in [2,9]. In this paper, we construct infinitely many new (basic) minimal forbidden configurations and thus obtain the answer this problem. We also obtain several necessary conditions for minimal forbidden configurations and give a generalization of Thomassens Theorem.