We define an infinite set of families of graphs, which we call $p$-wheels and denote ${\mathrm{}\left(\mathrm{Wh}\right)\mathrm{}}_n^{\mathrm{}\left(p\right)\mathrm{}}$, that generalize the wheel $\mathrm{}(p=1\mathrm{})\mathrm{}$ and biwheel $\mathrm{}(p=2\mathrm{})\mathrm{}$ graphs. The chromatic polynomial for ${\mathrm{}\left(\mathrm{Wh}\right)\mathrm{}}_n^{\mathrm{}\left(p\right)\mathrm{}}$ is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at $q=0,1,\dots ,p+1\mathrm{}$ for $n-p$ even and $q=0,1,\dots ,p+2\mathrm{}$ for $n-p$ odd; and (ii) the complex zeros all lie, equally spaced, on the unit circle $|q-\mathrm{}(p+1\mathrm{})\mathrm{}|=1\mathrm{}$ 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 $W(\{{\mathrm{}\left(\mathrm{Wh}\right)\mathrm{}}^{\mathrm{}\left(p\right)\mathrm{}}\},q)$ is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted.

• Received 12 March 1997

DOI:https://doi.org/10.1103/PhysRevE.56.1342