Are sparse graphs typically determined by their spectrum?

TL;DR

This paper explores whether sparse graphs are typically uniquely identified by their spectra, showing that certain components can be cospectral and proposing methods to understand when spectra determine graph structure.

Contribution

It introduces new results on cospectrality of sparse graphs, especially the giant component of Erdős-Rényi graphs, and analyzes conditions under which spectra determine graph structure.

Findings

Giant component of sparse Erdős-Rényi graphs can be cospectral.

Pendant trees are key obstructions to spectral uniqueness.

Local switching methods do not cause the 2-core to be cospectral.

Abstract

We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erd\H{o}s-R\'enyi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.