A note on unimodular congruence of graphs (Q1329996)

Let \(Q= Q(G)\) be an oriented vertex-edge incidence matrix of the graph \(G\). Then \(L(G)= QQ^ t\) is the Laplacian matrix of \(G\) and \(K(G)= Q^ t Q\) is an edge version of the Laplacian. In this note it is shown that for connected graphs \(G\), \(H\), associated matrices \(K(G)\) and \(K(H)\), respectively, are unimodularly congruent if and only if \(G\) and \(H\) have the same number of vertices and the same number of edges. The related problem of unimodular congruence of Laplacian matrices has been recently solved by \textit{W. Watkins} [Unimodular congruence of the Laplacian matrix of a graph, Linear Algebra Appl. 201, 43-49 (1994; next review)].