On cospectral graphons

TL;DR

This paper introduces the concept of cospectral graphons, providing multiple equivalent definitions and an example illustrating their unique properties compared to cospectral graphs.

Contribution

It extends the notion of cospectrality from graphs to graphons, offering new theoretical insights and an example of their limitations.

Findings

Defined cospectral graphons with three equivalent characterizations

Provided an example of non-approximability by cospectral graphs

Extended spectral graph theory to the graphon setting

Abstract

In this short note, we introduce cospectral graphons, paralleling the notion of cospectral graphs. As in the graph case, we give three equivalent definitions: by equality of spectra, by equality of cycle densities, and by a unitary transformation. We also give an example of two cospectral graphons that cannot be approximated by two sequences of cospectral graphs in the cut distance.