Constructing Cospectral Vertices Through Orbits of Subgraphs

TL;DR

This paper presents a method to construct cospectral vertices in undirected graphs, extending to the graph Laplacian, with conditions for strong cospectrality and an operation that preserves these properties.

Contribution

It introduces a new constructive approach for generating cospectral and strongly cospectral vertices, generalizing to the graph Laplacian and providing preservation operations.

Findings

Constructs cospectral vertices via a new method.

Extends the construction to the graph Laplacian.

Provides conditions for strong cospectrality.

Abstract

A constructive method is given for obtaining cospectral vertices in undirected graphs, along with an operation that preserves this construction. We prove that the construction yields cospectral vertices, as well as strongly cospectral vertices under additional conditions. Furthermore, we generalize cospectral vertices to the case of the graph Laplacian and provide an analogous construction.