A sufficient condition for generalized spectral characterization of graphs with loops
Combinatorics
2026-02-25 v2 Number Theory
Spectral Theory
Abstract
Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.
Cite
@article{arxiv.2511.19625,
title = {A sufficient condition for generalized spectral characterization of graphs with loops},
author = {Alexander Van Werde},
journal= {arXiv preprint arXiv:2511.19625},
year = {2026}
}
Comments
Added conjecture on satisfaction frequency