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

@article{Shrock1997FamiliesOG,
  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},
  year={1997},
  volume={56},
  pages={3935-3943}
}
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… 

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