Geodesic
High-ordered spectral characterization of unicyclic graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 1107-1141

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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 G^m be the m-th power (hypergraph) of G. The spectrum of G is referring to its adjacency matrix, and the spectrum of G^m 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 H^m is cospectral with G^m 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.
Keywords: unicyclic graph, graph isomorphism, cospectral graphs, power hypergraph, adjacency tensor, trace 
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Fan, Yi-Zheng; Yang, Hong-Xia; Zheng, Jian. High-ordered spectral characterization of unicyclic graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 1107-1141. http://geodesic.mathdoc.fr/item/DMGT_2024_44_3_a15/