1,239 research outputs found
Some inequalities for the Tutte polynomial
We prove that the Tutte polynomial of a coloopless paving matroid is convex
along the portions of the line segments x+y=p lying in the positive quadrant.
Every coloopless paving matroids is in the class of matroids which contain two
disjoint bases or whose ground set is the union of two bases of M*. For this
latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a
>= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same
class of matroids. We also prove this conjecture for some families of graphs
and matroids.Comment: 17 page
The algebra of flows in graphs
We define a contravariant functor K from the category of finite graphs and
graph morphisms to the category of finitely generated graded abelian groups and
homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j,
an element of Hom(K^j(X),B) is a coherent family of B-valued flows on the set
of all graphs obtained by contracting some (j-1)-set of edges of X; in
particular, Hom(K^1(X),R) is the familiar (real) ``cycle-space'' of X. We show
that K(X) is torsion-free and that its Poincare polynomial is the
specialization t^{n-k}T_X(1/t,1+t) of the Tutte polynomial of X (here X has n
vertices and k components). Functoriality of K induces a functorial coalgebra
structure on K(X); dualizing, for any ring B we obtain a functorial B-algebra
structure on Hom(K(X),B). When B is commutative we present this algebra as a
quotient of a divided power algebra, leading to some interesting inequalities
on the coefficients of the above Poincare polynomial. We also provide a formula
for the theta function of the lattice of integer-valued flows in X, and
conclude with ten open problems.Comment: 31 pages, 1 figur
Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings
We present exact results on the partition function of the -state Potts
model on various families of graphs in a generalized external magnetic
field that favors or disfavors spin values in a subset of
the total set of possible spin values, , where and are
temperature- and field-dependent Boltzmann variables. We remark on differences
in thermodynamic behavior between our model with a generalized external
magnetic field and the Potts model with a conventional magnetic field that
favors or disfavors a single spin value. Exact results are also given for the
interesting special case of the zero-temperature Potts antiferromagnet,
corresponding to a set-weighted chromatic polynomial that counts
the number of colorings of the vertices of subject to the condition that
colors of adjacent vertices are different, with a weighting that favors or
disfavors colors in the interval . We derive powerful new upper and lower
bounds on for the ferromagnetic case in terms of zero-field
Potts partition functions with certain transformed arguments. We also prove
general inequalities for on different families of tree graphs.
As part of our analysis, we elucidate how the field-dependent Potts partition
function and weighted-set chromatic polynomial distinguish, respectively,
between Tutte-equivalent and chromatically equivalent pairs of graphs.Comment: 39 pages, 1 figur
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