It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is...

Recently some important results have been proved showing that the gap between the largest eigenvalue A: of a finite regular graph of valency k and its second eigenvalue is related to expansion properties of the graph [1]. In this paper we investigate infinite graphs and show that in this case the expansion properties are related to the spectral radius of the graph. First we introduce necessary ...

Let be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ − cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with . Using the minimal polynomial of A and the theory of Smith normal forms we first determine which p-ranks of A follow directly from the ...

The spectrum of a Johnson graph is known to be given by the Eberlein polynomial. In this paper, a straightforward representation-theoretic derivation of this fact is presented. Also discussed are some consequences of this formula, such as the fact that infinitely many of them are Ramanujan. 1. Introduction. Johnson graphs are heavily studied objects. The spectrum of any graph—that is, the multi...

In this thesis we discuss some new results concerning the combinatorial Laplace operator of a simplicial complex. The combinatorial Laplacian of a simplicial complex encodes information about the relationships between adjacent simplices in the complex. This thesis is divided into two relatively disjoint parts. In the first portion of the thesis, we derive a relationship between the Laplacian sp...