Degree similar and net-degree similar signed graphs

Abstract

Two graphs are degree similar if their adjacency matrices and their degree matrices are simultaneously similar, a notion recently introduced by Godsil and Sun. We extend it to signed graphs in two ways, degree similarity and net-degree similarity, according to whether the degree or the net-degree matrix is used. We prove that signed trees are determined by their degree similarity class, but not by their net-degree similarity class. We then develop constructions of (net-)degree similar signed graphs, based on a signed version of Godsil--McKay switching, on partial transposition and on block extensions. As a byproduct, we obtain infinite families both of pairs of net-degree similar unicyclic signed graphs with non-isomorphic underlying graphs, and of unicyclic signed graphs, not necessarily regular, which are not determined by their degree similarity class. This settles the net-degree version of a problem of Godsil and Sun on unicyclic graphs.