High-ordered spectral characterization of unicyclic graphs

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let G be a graph and Gm be the m-th power (hypergraph) of G. The spectrum of G is referring to its adjacency matrix, and the spectrum of Gm is referring to its adjacency tensor. The graph G is called determined by high-ordered spectra (DHS, for short) if, whenever H is a graph such that Hm is cospectral with Gm for all m, then H is isomorphic to G. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.