Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0

  title={Families of Graphs with W\_r(\{G\},q) Functions That Are Nonanalytic at 1/q=0},
  author={Robert Shrock and Shan-Ho Tsai},
  journal={Physical Review E},
Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the following: is $W_r(\{G\},q) = q^{-1}W(\{G\},q)$ analytic or not at the origin of the $1/q$ plane? (where the complex generalization of $q$ is assumed). This question is also relevant in statistical mechanics because $W(\{G\},q)=\exp(S_0/k_B)$, where $S_0$ is the… 

Figures from this paper

Chromatic Roots are Dense in the Whole Complex Plane

  • A. Sokal
  • Mathematics
    Combinatorics, Probability and Computing
  • 2004
The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

Lower bounds and series for the ground-state entropy of the Potts antiferromagnet on Archimedean lattices and their duals

We prove a general rigorous lower bound for $W(\Lambda,q)=\exp(S_0(\Lambda,q)/k_B)$, the exponent of the ground state entropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedean

Bounds on the Complex Zeros of ( Di ) Chromatic Polynomials and Potts-Model Partition Functions

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree 6 r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc |q| <

Chromatic polynomials of planar triangulations, the Tutte upper bound and chromatic zeros

  • R. ShrockYan Xu
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2012
Tutte proved that if Gpt is a planar triangulation and P(Gpt, q) is its chromatic polynomial, then |P(Gpt, τ + 1)| ⩽ (τ − 1)n − 5, where and n is the number of vertices in Gpt. Here we study the



Asymptotic limits and zeros of chromatic polynomials and ground-state entropy of Potts antiferromagnets

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in

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 $(Wh)^{(p)}_n$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for

Ground-state entropy of Potts antiferromagnets: Bounds, series, and Monte Carlo measurements

This paper reports several results concerning the exponent of the ground-state entropy of the Potts antiferromagnet on a lattice, and improves the previous rigorous lower bound on W(\mathrm{hc},q) for the honeycomb (hc) lattice and finds that it is extremely accurate.


We derive rigorous upper and lower bounds for the ground state entropy of the q-state Potts antiferromagnet on the honeycomb and triangular lattices. These bounds are quite restrictive, especially

Graph theory

Algebraic Graph Theory”2nd ed

  • 1993

See also J. F. Nagle

  • J. Combin. Theory B
  • 1968

See also the related work in N. L. Biggs

  • Bull. London Math. Soc
  • 1975

Theory B 10

  • 42 (1971). See also J. F. Nagle, J. Math. Phys. 9, 1007
  • 1968

) . See also the related work in N . L . Biggs , J . Phys . A 8 , L 110 ( 1975 )

  • Regular Polytopes