In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. ...

We describe three weight systems arising from the intersection graphs of chord diagrams (also known as circle graphs). These derive from the determinant and rank of the adjacency matrix of the graph, viewed over Z 2 , and from the rank of a " marked " adjacency matrix. We show that these weight systems are exactly those given by the Conway, HOMFLYPT and Kauffman polynomials, respectively.

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and fo...

This paper shows how the eigenstructure of the adjacency matrix can be used for the purposes of robust graph-matching. We commence from the observation that the leading eigenvector of a transition probability matrix is the steady state of the associated Markov chain. When the transition matrix is the normalised adjacency matrix of a graph, then the leading eigenvector gives the sequence of node...

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.

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d+1 distinct eigenvalues.

We consider the problem of discovering links of an evolving undirected graph given a series of past snapshots of that graph. The graph is observed through the time sequence of its adjacency matrix and only the presence of edges is observed. The absence of an edge on a certain snapshot cannot be distinguished from a missing entry in the adjacency matrix. Additional information can be provided by...

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...

It is well known that the smallest eigenvalue of the adjacency matrix of a connected ¿-regular graph is at least -d and is strictly greater than -d if the graph is not bipartite. More generally, for any connected graph G = (V JE), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G , and A is the adjacency matrix of G . Then Q is positive semi-definite, and th...