# Characterizing cospectral vertices via isospectral reduction

## TL;DR

This paper establishes a fundamental link between cospectral vertices and isospectral reductions, showing that cospectrality corresponds to symmetry in reduced graphs and providing new ways to construct graphs with cospectral vertices.

## Contribution

It proves that cospectral vertices correspond to automorphisms in isospectral reductions and characterizes strongly cospectral vertices through eigenvalue simplicity, extending prior symmetry results.

## Key findings

- Cospectral vertices correspond to automorphisms in isospectral reductions.
- Strongly cospectral vertices are characterized by simple eigenvalues in reductions.
- New graph families with cospectral vertices can be constructed using these results.

## Abstract

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices.