A note on 2-isomorphisms and the signed Laplacian matrix of a graph

Let G and H be graphs with a bijection tau on their edges. In a recent paper, we observed that the Laplacian matrices of signed versions of G and H contain enough information to decide whether or not G and H are dual graphs with respect to tau. In this note we show that G and H are 2-isomorphic if and only if the determinants of the reduced forms of the consistently signed Laplacian matrices for G and H are equal. Thus the Laplacian matrices of signed versions of G and H also contain enough information to decide whether or not G and H are 2-isomorphic with respect to tau. (C) 2018 Elsevier Inc. All rights reserved.