308 research outputs found
Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets
We study the asymptotic limiting function , where is the chromatic polynomial for a graph
with vertices. We first discuss a subtlety in the definition of
resulting from the fact that at certain special points , the
following limits do not commute: . We then
present exact calculations of and determine the corresponding
analytic structure in the complex plane for a number of families of graphs
, including circuits, wheels, biwheels, bipyramids, and (cyclic and
twisted) ladders. We study the zeros of the corresponding chromatic polynomials
and prove a theorem that for certain families of graphs, all but a finite
number of the zeros lie exactly on a unit circle, whose position depends on the
family. Using the connection of with the zero-temperature Potts
antiferromagnet, we derive a theorem concerning the maximal finite real point
of non-analyticity in , denoted and apply this theorem to
deduce that and for the square and
honeycomb lattices. Finally, numerical calculations of and
are presented and compared with series expansions and bounds.Comment: 33 pages, Latex, 5 postscript figures, published version; includes
further comments on large-q serie
Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
We calculate the chromatic polynomials and, from these, the
asymptotic limiting functions
for families of -vertex graphs comprised of repeated subgraphs
adjoined to an initial graph . These calculations of for
infinitely long strips of varying widths yield important insights into
properties of for two-dimensional lattices . In turn,
these results connect with statistical mechanics, since is the
ground state degeneracy of the -state Potts model on the lattice .
For our calculations, we develop and use a generating function method, which
enables us to determine both the chromatic polynomials of finite strip graphs
and the resultant function in the limit . From
this, we obtain the exact continuous locus of points where
is nonanalytic in the complex plane. This locus is shown to
consist of arcs which do not separate the plane into disconnected regions.
Zeros of chromatic polynomials are computed for finite strips and compared with
the exact locus of singularities . We find that as the width of the
infinitely long strips is increased, the arcs comprising elongate
and move toward each other, which enables one to understand the origin of
closed regions that result for the (infinite) 2D lattice.Comment: 48 pages, Latex, 12 encapsulated postscript figures, to appear in
Physica
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
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