Spectral characterizations of tournaments

Abstract

Characterizing graphs by the spectra of various matrices associated with the graphs has long been an important topic in spectral graph theory. However, it is generally very hard to show a given graph to be determined by its spectrum. This paper gives a simple criterion for a tournament to be determined by its adjacency spectrum. More precisely, let G be a tournament of order n with adjacency matrix A, and WA(G)=[e,Ae,...,An−1e] be its walk matrix, where e is the all-one vector. We show that, for any self-converse tournament G, if det⁡WA(G) is square-free, then G is uniquely determined by its adjacency spectrum among all tournaments. The result is somewhat unexpected, which was achieved by employing some recent tools developed by the second author for showing a graph to be determined by its generalized spectrum. The key observation is that for a tournament G, the complement of G coincides with the converse of G, and hence a certain equivalence between the ordinary adjacency spectrum and the generalized (skew)-adjacency spectrum for G can be established. Moreover, some equivalent formulations of the above result, as well as some related ones are also presented.