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Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertexgraph $G$ 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 thefollowing: is $W_r(\{G\},q) = q^{-1}W(\{G\},q)$ analytic or not at the originof the $1/q$ plane? (where the complex generalization of $q$ is assumed). Thisquestion is also relevant in statistical mechanics because$W(\{G\},q)=\exp(S_0/k_B)$, where $S_0$ is the ground state entropy of the$q$-state Potts antiferromagnet on the lattice graph $\{G\}$, and theanalyticity of $W_r(\{G\},q)$ at $1/q=0$ is necessary for the large-$q$ seriesexpansions of $W_r(\{G\},q)$. Although $W_r$ is analytic at $1/q=0$ for many$\{G\}$, there are some $\{G\}$ for which it is not; for these, $W_r$ has nolarge-$q$ series expansion. It is important to understand the reason for thisnonanalyticity. Here we give a general condition that determines whether or nota particular $W_r(\{G\},q)$ is analytic at $1/q=0$ and explains thenonanalyticity where it occurs. We also construct infinite families of graphswith $W_r$ functions that are non-analytic at $1/q=0$ and investigate theproperties of these functions. Our results are consistent with the conjecturethat a sufficient condition for $W_r(\{G\},q)$ to be analytic at $1/q=0$ isthat $\{G\}$ is a regular lattice graph $\Lambda$. (This is known not to be anecessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in  Phys. Rev.

physical review. e, statistical physics, plasmas, fluids, and related interdisciplinary topics

Families of graphs with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>{</mml:mi><mml:mi>G</mml:mi><mml:mi>}</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:math>functions that are nonanalytic at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>/</mml:mo><…

Families of graphs withWr({G},q)functions that are nonanalytic at1/<…