The Degree Polynomial

TL;DR

This work studies the zeros of degree polynomials $D(G;x)=\sum_{v} x^{\deg(v)}$ associated with graphs and multigraphs, aiming to understand where these roots lie and how they relate to degree sequences. By analyzing two representative families— CL_n (complete graphs with a leaf) and anti-regular graphs $H_n$ and $H_n^c$—the authors apply tools such as Descartes' rule of signs, the Intermediate Value Theorem, Rouche's theorem, Eneström-Kakeya bounds, and Beraha-Kahane-Weiss theory to characterize root locations and accumulation sets. They establish concrete patterns, including a dominant large negative root for CL_n and a unit-circle cluster of other roots, explicit root structures for even/odd anti-regular graphs, and general modulus bounds with extremal cases precisely described. A conjectured region $|z|\le (n-1)^{|\arg(z)|/\pi}$ is proposed to tightly bound roots across graph families, inviting further exploration of broader graph classes and rational roots.

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 $n$.

Figures (6)

• Figure 1: Degree roots of graphs of small order $n$.
• Figure 2: All roots of $D(CL_n;x)$ for $2 \leq n \leq 20$.
• Figure 3: The roots of $D(CL_n;x)$ for $2 \leq n \leq 50$ that are contained in the unit circle. The roots appear to be converging outward to the unit circle as $n$ increases.
• Figure 4: Examples of anti-regular graphs and their degree polynomials. Left to right: $H_2, H_3, H_4, H_5$.
• Figure 5: Degree roots for connected anti-regular graphs $H_n$, up to order $n=50$ (red roots correspond to even $n$, blue to odd $n$.

Theorems & Definitions (16)

• Proposition 2.1
• proof
• Theorem 2.2: Beraha-Kahane-Weiss, bkw1975
• Theorem 2.3: brown2020extension
• Proposition 2.4
• Theorem 2.5: Eneström-Kakeya enestrom1920remarquekakeya1912limits
• Proposition 2.6
• Proposition 3.1
• Theorem 3.2
• Proposition 3.3