Singularly cospectral circulant graphs

TL;DR

This paper investigates the spectral properties of circulant graphs, establishing conditions for singular cospectrality and demonstrating isomorphism in certain cases with odd prime vertices.

Contribution

It provides new sufficient conditions for noncospectral singularly cospectral circulant graphs and proves isomorphism for such graphs with an odd prime number of vertices.

Findings

Sufficient conditions for noncospectral singularly cospectral circulant graphs.

Analysis of inertia in these graphs.

Proof that such graphs with odd prime vertices are isomorphic.

Abstract

Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse may be false. In this paper, we present sufficient conditions for two circulant graphs, with an even number of vertices, to be noncospectral singularly cospectral graphs. In this analysis, we study when a pair of these graphs have the same or distinct inertia. In addition, we show that two singularly cospectral circulant graphs with an odd prime number of vertices are isomorphic.