Clique complexes of strongly regular graphs, their eigenvalues, and cohomology groups

TL;DR

This paper explores whether higher order Laplacian spectra of clique complexes can differentiate strongly regular graphs with identical parameters, revealing cases where spectra distinguish graphs and cases where they do not, and analyzing cohomology groups.

Contribution

It investigates the spectra of higher order Laplacians of clique complexes for strongly regular graphs and establishes conditions for trivial cohomology groups.

Findings

Spectra of triangle complexes can distinguish many strongly regular graphs with the same parameters.

Some strongly regular graphs share identical higher order Laplacian spectra, making them indistinguishable by this method.

Graphs with certain induced cycle properties have trivial first cohomology groups.

Abstract

It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can distinguish them. In this direction, we study the clique complexes of strongly regular graphs, and determine the spectra of the triangle complexes of several families of strongly regular graphs including Hamming graphs and Triangular graphs. In many cases, the spectrum of the triangle complex distinguishes between strongly regular graphs with the same parameters, but we find some examples where that is not the case. We also prove that if a graph has the property that for any induced cycle, there are four consecutive vertices on the cycle with a common neighbor, then the first cohomology group of the graph is trivial and we apply this result to several families of graphs.