A cospectral family of graphs for the normalized Laplacian found by toggling

TL;DR

This paper introduces a novel construction of weighted graphs that are pairwise cospectral with respect to the normalized Laplacian, enabling the creation of graph pairs with differing edge counts, including subgraph relationships.

Contribution

It presents a new method for constructing cospectral graphs for the normalized Laplacian using toggling, expanding the understanding of spectral graph properties.

Findings

Constructed a family of cospectral weighted graphs

Demonstrated cospectrality via equal characteristic polynomials

Allowed for graphs with different numbers of edges, including subgraphs

Abstract

We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral graphs with differing number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.