Families of Graphs With Chromatic Zeros Lying on Circles

TL;DR

This paper introduces a new family of graphs called p-wheels, calculates their chromatic polynomials, and reveals that their chromatic zeros lie on specific circles in the complex plane, with implications for graph theory and statistical mechanics.

Contribution

The paper defines p-wheels, derives their chromatic polynomials, and uncovers the geometric distribution of their zeros, extending understanding of chromatic zeros on circles.

Findings

Real zeros at q=0,1,...,p+1 or p+2 depending on parity

Complex zeros evenly spaced on a circle |q-(p+1)|=1

Zeros form a boundary curve in the limit as n approaches infinity

Abstract

We define an infinite set of families of graphs, which we call $p$-wheels and denote $(Wh)^{(p)}_n$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for $(Wh)^{(p)}_n$ is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at $q=0,1,...p+1$ for $n-p$ even and $q=0,1,...p+2$ for $n-p$ odd; and (ii) the complex zeros all lie, equally spaced, on the unit circle $|q-(p+1)|=1$ in the complex $q$ plane. In the $n \to \infty$ limit, the zeros on this circle merge to form a boundary curve separating two regions where the limiting function $W(\{(Wh)^{(p)}\},q)$ is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted.