2,523 research outputs found
Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions
Counting problems, determining the number of possible states of a large
system under certain constraints, play an important role in many areas of
science. They naturally arise for complex disordered systems in physics and
chemistry, in mathematical graph theory, and in computer science. Counting
problems, however, are among the hardest problems to access computationally.
Here, we suggest a novel method to access a benchmark counting problem, finding
chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern
matching algorithm that exploits the equivalence between the chromatic
polynomial and the zero-temperature partition function of the Potts
antiferromagnet on the same graph. Implementing this bottom-up algorithm using
appropriate computer algebra, the new method outperforms standard top-down
methods by several orders of magnitude, already for moderately sized graphs. As
a first application, we compute chromatic polynomials of samples of the simple
cubic lattice, for the first time computationally accessing three-dimensional
lattices of physical relevance. The method offers straightforward
generalizations to several other counting problems.Comment: 7 pages, 4 figure
Euler characteristic reciprocity for chromatic, flow and order polynomials
The Euler characteristic of a semialgebraic set can be considered as a
generalization of the cardinality of a finite set. An advantage of
semialgebraic sets is that we can define "negative sets" to be the sets with
negative Euler characteristics. Applying this idea to posets, we introduce the
notion of semialgebraic posets. Using "negative posets", we establish Stanley's
reciprocity theorems for order polynomials at the level of Euler
characteristics. We also formulate the Euler characteristic reciprocities for
chromatic and flow polynomials.Comment: 16 pages; flow polynomial reciprocity added; title change
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
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
A bivariate chromatic polynomial for signed graphs
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial
which counts all -colorings of a graph such
that adjacent vertices get different colors if they are . Our first
contribution is an extension of to signed graphs, for which we
obtain an inclusion--exclusion formula and several special evaluations giving
rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is
to derive combinatorial reciprocity theorems for and its
signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking
chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure
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