The Degree Polynomial

TL;DR

This paper studies the roots of degree polynomials of multigraphs, exploring their properties, bounds on their moduli, and regions containing all roots for graphs of a given size.

Contribution

It introduces bounds and a region that contains all roots of degree polynomials for multigraphs, advancing understanding of their root distributions.

Findings

Bounds on the moduli of roots established

A region containing all roots for multigraphs of order n proposed

Properties of roots for specific graph families analyzed

Abstract

The degree polynomial of a multigraph $G$ is given by $\sum _{v \in V(G)} x^{\mbox{deg}(v)}$. We investigate here properties of the roots of such polynomials. In addition to examining the roots for some families of graphs with few and many degrees, we provide some bounds on the moduli of the roots. We also propose a region that contains all roots for multigraphs of order $n$.