Oriented graphs determined by their generalized skew spectrum

Spectral characterization of graphs is a well-studied topic in spectral graph theory which has received a lot of attention from researchers. The spectral characterization of oriented graphs, however, is less studied so far. Given a simple undirected graph G with an orientation sigma, the oriented graph G(sigma) is a digraph obtained from G by assigning to every edge of G a direction according to sigma. An oriented graph G(sigma) is self-converse if it is isomorphic to its converse (G(sigma))(T), a graph obtained from G(sigma) by reversing each directed edge in G(sigma). We denote by S(G(sigma)) the skew-adjacency matrix of G(sigma). For two oriented graphs G(sigma) and H-tau with skew-adjacency matrices S(G(sigma)) and S(H-tau), respectively, we say G(sigma) is R-cospectral to H-tau, if for any t is an element of R, two matrices tJ - S(G(sigma)) and tJ - S(H-tau) have the same spectrum, where J is the all-one matrix. An oriented graph G(sigma) is said to be determined by the generalized skew spectrum (DGSS for short) if, any oriented graph R-cospectral to G(sigma) is isomorphic to G(sigma). Oriented graphs that are DGSS must be self-converse. In this paper, we give a simple arithmetic condition for a self-converse oriented graph being DGSS, which provides an analogue of a similar result for ordinary graphs; see[15]. More precisely, let G(sigma) be a self-converse oriented graph with skew-adjacency matrix S(G(sigma)), and W(G(sigma)) =[e, S(G(sigma)) e, ..., Sn-1 (G(sigma))e](e is the all-one vector) be the skew-walk-matrix. We show that if 2(-[n/2]) det W(G(sigma)) is odd and square-free, then G(sigma) is DGSS. Moreover, we also illustrate that our result is the best possible in certain sense. (c) 2021 Elsevier Inc. All rights reserved.