Families of Graphs With Chromatic Zeros Lying on Circles

Abstract

We define an infinite set of families of graphs, which we call $p$-wheels and denote $(Wh{)}_n^{(p)}$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for $(Wh{)}_n^{(p)}$ 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.

8 pages, Latex

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