Constructing Non-isomorphic Signless Laplacian Cospectral Graphs

TL;DR

This paper introduces a method using partial transpose to generate large families of non-isomorphic graphs that share the same signless Laplacian spectrum, significantly advancing graph spectral theory.

Contribution

It presents a novel construction technique for non-isomorphic signless Laplacian cospectral graphs using partial transpose, with high efficiency for small graphs.

Findings

Over 70% of such graphs with up to 8 vertices can be generated

Numerous examples of cospectral non-isomorphic graphs are provided

The method is significantly powerful for constructing cospectral graphs

Abstract

In this article, we generate large families of non-isomorphic and signless Lalacian cospectral graphs using partial transpose on graphs. Our constructions are significantly powerful. More than $70\%$ of non-isomorphic signless-Laplacian cospectral graphs can be generated with partial transpose when number of vertices is $\le 8$. We have also produced numerous examples of non-isomorphic signless Laplacian cospectral graphs.