5 research outputs found
Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs
A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added
references, clarified examples. 35 p
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
Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure