Families of Graphs With Chromatic Zeros Lying on Circles

Details Families of Graphs With Chromatic Zeros Lying on Circles Families of Graphs With Chromatic Zeros Lying on Circles

1997-03-28

arxiv.org/abs/cond-mat/9703249v1

Phys. Rev. E56, 1342 (1997)

Physics

Condensed Matter

Statistical Mechanics

8 pages, Latex

Scientific paper

10.1103/PhysRevE.56.1342

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.

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