Families of Graphs With Chromatic Zeros Lying on Circles

We define an infinite set of families of graphs, which we call p-wheels and denote (Wh) n , that generalize the wheel (p = 1) and biwheel (p = 2) graphs. The chromatic polynomial for (Wh) 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 → ∞ limit, the zeros on this circle merge to form a boundary curve separating two regions where the limiting function W ({(Wh)}, q) is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted. ∗email: [email protected] ∗∗email: [email protected]