A construction of cospectral graphs for the normalized Laplacian

TL;DR

This paper introduces a method for constructing large families of non-bipartite, non-regular graphs that are cospectral with respect to the normalized Laplacian, using local modifications involving bipartite subgraphs.

Contribution

It presents a novel local modification technique to generate cospectral graphs for the normalized Laplacian, expanding known classes of such graphs.

Findings

Produced exponentially large families of cospectral graphs

Demonstrated cospectrality with complements without self-complementarity

Provided a method for replacing bipartite subgraphs with cospectral mates

Abstract

We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix $A+tD$. We produce (exponentially) large families of non-bipartite, non-regular graphs which are mutually cospectral, and also give an example of a graph which is cospectral with its complement but is not self-complementary.