Finding singularly cospectral graphs

TL;DR

This paper investigates the properties and conditions of singularly cospectral graphs, providing necessary and sufficient criteria, and constructs infinite families of such graphs with specific spectral characteristics.

Contribution

It offers a complete characterization of singularly cospectral graphs and constructs infinite examples, addressing a problem posed by Nikiforov.

Findings

Necessary and sufficient conditions for singularly cospectral graphs

Construction of infinite families of noncospectral singularly cospectral graphs

Analysis of the relationship between almost cospectrality and singularly cospectrality

Abstract

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree.