Isospectral discrete and quantum graphs with the same flip counts and nodal counts

TL;DR

This paper presents a new method for constructing pairs of isospectral discrete and quantum graphs that share the same flip and nodal counts, addressing the challenge of distinguishing non-isomorphic graphs with identical spectra.

Contribution

It introduces a simple alternative mechanism to generate systematic examples of isospectral graphs with identical flip and nodal counts, expanding understanding of spectral graph properties.

Findings

Constructed examples of isospectral graphs with identical flip and nodal counts

Demonstrated a new mechanism for producing such graphs systematically

Extended the class of known isospectral graphs with shared spectral and nodal properties

Abstract

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count). Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay switching. Here we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts.