Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates

TL;DR

This paper introduces a new upper bound on the number of non-isomorphic generalized cospectral mates of a graph, based on the walk matrix's determinant, advancing understanding in spectral graph theory.

Contribution

It establishes a novel upper bound on generalized cospectral mates using walk matrix properties, providing a refined arithmetic criterion for their multiplicity.

Findings

Derived an explicit upper bound based on walk matrix determinant

Connected the bound to the arithmetic structure of the walk matrix

Enhanced understanding of the structure and multiplicity of generalized cospectral graphs

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.