On the number of generalized cospectral mates of graphs

TL;DR

This paper derives a tight upper bound on the number of non-isomorphic graphs sharing the same generalized spectrum, broadening understanding of spectral graph uniqueness using Smith Normal Form constraints.

Contribution

It introduces a novel arithmetic approach based on Smith Normal Form to bound the number of generalized cospectral mates, extending spectral uniqueness results to more graph classes.

Findings

Established a tight upper bound on generalized cospectral mates

Extended spectral uniqueness to broader graph classes

Applied Smith Normal Form to spectral graph analysis

Abstract

This paper establishes an upper bound on the number of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectrum of a graph and its complement. Moving beyond the classical problem of identifying graphs determined by their generalized spectrum, we address the more quantitative question of how many non-isomorphic graphs can share the same generalized spectrum. Our approach is based on arithmetic constraints derived from the Smith Normal Form (SNF) of the walk matrix, which leads to a tight upper bound on the number of generalized cospectral mates of a graph. Our upper bound applies to a much broader class of graphs than those previously shown to have no generalized cospectral mates (graphs determined by generalized spectrum). Consequently, this work extends the family of graphs for which strong and informative spectral uniqueness results are available