# Polynomial Relations Between Matrices of Graphs

## TL;DR

This paper explores polynomial relations between matrices associated with graphs, specifically between the adjacency matrix and the signless Laplacian, revealing that such relations mainly occur in regular or biregular graphs.

## Contribution

It establishes a connection between eigenvalues of key graph matrices and characterizes when polynomial relations between these matrices can exist.

## Key findings

- Polynomial relations between $A$ and $Q$ exist for biregular graphs.
- Such polynomial relations are essentially limited to regular or biregular graphs.
- The relation $A^2=(Q-d_1I)(Q-d_2I)$ links eigenvalues of $A$ and $Q$ in biregular graphs.

## Abstract

We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when it is possible to have $X^r=f(Y)$ for $f$ a polynomial, $r>0$, and $X,\ Y$ matrices associated to a graph $G$. It turns out that, essentially, this can only happen if $G$ is either regular or biregular.