8 research outputs found
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics
On the chromatic roots of generalized theta graphs
The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of
endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1.
We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z)
of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)]
k/\log k, uniformly in the path lengths s_i. Moreover, we prove that
\Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 +
o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph
with a chromatic root that maximizes |z-1| is the one with all path lengths
equal to 2; we conjecture that this holds for all k.Comment: LaTex2e, 25 pages including 2 figure
On the location of chromatic zeros of series-parallel graphs
In this paper we consider the zeros of the chromatic polynomial of
series-parallel graphs. Complementing a result of Sokal, showing density
outside the disk , we show density of
these zeros in the half plane and we show there exists an open
region containing the interval such that does
not contain zeros of the chromatic polynomial of series-parallel graphs.
We also disprove a conjecture of Sokal by showing that for each large enough
integer there exists a series-parallel graph for which all vertices
but one have degree at most and whose chromatic polynomial has a zero
with real part exceeding .Comment: 18 pages, 2 figure