We consider weighted graphs, where the edgeweights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained. © 2007 Elsevier B.V. All rights reserved.

We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn, n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there...

Let G be a simple graph with finite number of vertices. We denote by det(G) the determinant of an adjacency matrix of G. This number det(G) is an integer and is an invariant of G so that its value is independent of the choice of vertices in an adjacency matrix. In this paper, we study the distributions of det(G) whenever G runs over graphs with finite n vertices for a given integer n ≥ 1. We de...

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...

A block graph is a graph in which every block is a complete graph. Let G be a block graph and let A be the adjacency matrix of G. We first obtain a formula for the determinant of A over reals. It is shown that A is nonsingular over IF2 if and only if the removal of any vertex from G produces a graph with exactly one odd component. A formula for the inverse of A over IF2 is obtained, whenever it...

Abstract Inspired by the notion of action convergence in graph limit theory, we introduce a measure-theoretic representation matrices, and use it to define new pseudo-metric on space matrices. Moreover, show that such is metric subspace adjacency or Laplacian matrices for graphs. Hence, particular, obtain isomorphism classes Additionally, study how some properties graphs translate this measure ...