Using twins and scaling to construct cospectral graphs for the normalized Laplacian

TL;DR

This paper introduces a method using twins and scaling to construct pairs of cospectral graphs with different numbers of edges for the normalized Laplacian, addressing a longstanding challenge in spectral graph theory.

Contribution

It presents a novel construction technique for cospectral graphs with differing edge counts using twins and scaling properties.

Findings

Constructed cospectral graphs with different edge numbers

Provided examples of graphs cospectral with subgraphs

Demonstrated the use of twins and scaling in spectral graph construction

Abstract

The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Laplacian matrix.