Families of graphs with W_{r}(\{G\},q) functions that are nonanalytic at 1/q=0
Abstract
Denoting P(G,q) as the chromatic polynomial for coloring an nvertex graph G with q colors, and considering the limiting function W(\{G\},q)=_{n>∞}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 ground state entropy of the qstate Potts antiferromagnet on the lattice graph \{G\}, and the analyticity of W_{r}(\{G\},q) at 1/q=0 is necessary for the largeq series expansions 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 no largeq series expansion. It is important to understand the reason for this nonanalyticity. Here we give a general condition that determines whether or not a particular W_{r}(\{G\},q) is analytic at 1/q=0 and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with W_{r} functions that are nonanalytic at 1/q=0 and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for W_{r}(\{G\},q) to be analytic at 1/q=0 is that \{G\} is a regular lattice graph Λ. (This is known not to be a necessary condition.)
 Publication:

Physical Review E
 Pub Date:
 October 1997
 DOI:
 10.1103/PhysRevE.56.3935
 arXiv:
 arXiv:condmat/9707096
 Bibcode:
 1997PhRvE..56.3935S
 Keywords:

 05.20.y;
 64.60.Cn;
 75.10.Hk;
 Classical statistical mechanics;
 Orderdisorder transformations;
 statistical mechanics of model systems;
 Classical spin models;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Lattice;
 Mathematics  Combinatorics
 EPrint:
 22 pages, Revtex, 4 encapsulated postscript figures, to appear in Phys. Rev. E