Abstract

Let G be a simple undirected graph (no loops, no multiple edges) on n vertices. Let S n be the set of real symmetric matrices of order n. A matrix A = (a i, j) ∈ S n is said to be a matrix on G if a i, j = 0 whenever i ≠ j and the vertices i, j of G are not joined by an edge of G. We recall that if F is a skew-symmetric operator on S n, then the solution A(t) offenced((frac(d A, d t) = [A, F (A)]; A (0) = A 0 ∈ S n))maintains the spectrum of A 0. The matrix A ∈ S n is said to be centrosymmetric if JAJ = A, where J is the matrix with ones on the secondary diagonal and zeros elsewhere. Centrosymmetric matrices are symmetric about the secondary diagonal. Centrosymmetric matrices appear in fields such as finite element analysis. We construct an isospectral flow on a graph G, with the property that if A 0 is centrosymmetric, so is A(t), and discuss the limit of A(t) as t → ∞. © 2006 Elsevier Inc. All rights reserved.