Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates
Abstract
The problem of characterizing graphs determined by their spectrum (DS) or generalized spectrum (DGS) has been a longstanding topic of interest in spectral graph theory, originating from questions in chemistry and mathematical physics. While previous studies primarily focus on identifying whether a graph is DGS, we address a related yet distinct question: how many non-isomorphic generalized cospectral mates a graph can have? Building upon recent advances that connect this question to the properties of the walk matrix, we introduce a broad family of graphs and establish an explicit upper bound on the number of non-isomorphic generalized cospectral mates they can have. This bound is determined by the arithmetic structure of the determinant of the walk matrix, offering a refined criterion for quantifying the multiplicity of generalized cospectral graphs. This result sheds new light on the structure of generalized cospectral graphs and provides a refined arithmetic criterion for bounding their multiplicity.
Cite
@article{arxiv.2507.06927,
title = {Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates},
author = {Muhammad Raza and Mudassir Shabbir and Waseem Abbas},
journal= {arXiv preprint arXiv:2507.06927},
year = {2025}
}