Graphs with at most one generalized cospectral mate

TL;DR

This paper introduces a large family of graphs with specific spectral properties, proving most are uniquely determined by their spectra or have a unique cospectral mate, and develops algorithms to identify these cases efficiently.

Contribution

The paper defines the family _n, proves most graphs in it are DGS or have a unique cospectral mate, and creates an efficient method to determine this, supporting Haemers' conjecture.

Findings

Most graphs in _n are DGS or have a unique cospectral mate.

The problem of finding cospectral mates reduces to generating rational orthogonal matrices.

Experimental results suggest _n has positive density and most are DGS as n grows.

Abstract

Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor n/2 \rfloor}\det W$ is odd and square-free. In this paper, we introduce a large family of graphs $\mathcal{F}_n=\{$ $n$-vertex graphs $G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b$ and rank$W=n-1$ over $\mathbb{Z}/p\mathbb{Z}\},$ where $b$ is odd and square-free, $p$ is an odd prime and $p\nmid b$. We prove that any graph in $\mathcal{F}_n$ either is DGS or has exactly one generalized cospectral mate up to isomorphism. Moreover, we show that the problem of finding the generalized cospectral mate for a graph in $\mathcal{F}_n$ is equivalent to that of generating an appropriate rational orthogonal matrix from a given integral vector. This equivalence essentially depends on an amazing property of graphs in terms of generalized spectra, which states that any symmetric integral matrix generalized cospectral with the adjacency matrix of some graph must be an adjacency matrix. Based on this equivalence, we develop an efficient algorithm to decide whether a given graph in $\mathcal{F}_n$ is DGS and further to find the unique generalized cospectral mate when it is not. We give some experimental results on graphs with at most 20 vertices, which suggest that $\mathcal{F}_n$ may have a positive density (nearly $3\%$) and possibly almost all graphs in $\mathcal{F}_n$ are DGS as $n\rightarrow \infty$. This gives a supporting evidence for Haemers' conjecture that almost all graphs are determined by their spectra.