Unimodular Congruence Of The Laplacian Matrix Of A Graph

Let G be a graph with vertices 1, 2, …, n. Associated with G, there is an integral quadratic form, Q(x), on the n-tuple of indeterminates x= (x1, …, xn), given by Q(x) = Σ(xi− xj)2, where the sum is taken over all edges (i, j) of G. In this paper we prove that the quadratic forms Q1, Q2 associated with graphs G1, G2 are congruent by a unimodular matrix if and only if G1 and G2 are cycle isomorphic.