Isospectral graphs with identical nodal counts

TL;DR

This paper constructs the first non-trivial counter-examples to the conjecture that isospectral objects have different nodal count sequences, revealing a surprising link between isospectral discrete and quantum graphs.

Contribution

It introduces new counter-examples to a conjecture about isospectral objects and uncovers a surprising connection between discrete and quantum graphs.

Findings

Counter-examples to the conjecture were constructed.

A surprising connection between isospectral discrete and quantum graphs was demonstrated.

The results challenge previous assumptions about nodal counts and spectral properties.

Abstract

According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs.