14,088 research outputs found
Rank three matroids are Rayleigh
A Rayleigh matroid is one which satisfies a set of inequalities analogous to
the Rayleigh monotonicity property of linear resistive electrical networks. We
show that every matroid of rank three satisfies these inequalities.Comment: 11 pages, 3 figures, 3 table
Algebras related to matroids represented in characteristic zero
Let k be a field of characteristic zero. We consider graded subalgebras A of
k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear
forms. Representations of matroids over k provide a natural description of the
structure of these algebras. In return, the numerical properties of the Hilbert
function of A yield some information about the Tutte polynomial of the
corresponding matroid. Isomorphism classes of these algebras correspond to
equivalence classes of hyperplane arrangements under the action of the general
linear group.Comment: 11 pages AMS-LaTe
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
The Tutte dichromate and Whitney homology of matroids
We consider a specialization of the Tutte polynomial of a matroid
which is inspired by analogy with the Potts model from statistical
mechanics. The only information lost in this specialization is the number of
loops of . We show that the coefficients of are very simply
related to the ranks of the Whitney homology groups of the opposite partial
orders of the independent set complexes of the duals of the truncations of .
In particular, we obtain a new homological interpretation for the coefficients
of the characteristic polynomial of a matroid
On the imaginary parts of chromatic root
While much attention has been directed to the maximum modulus and maximum
real part of chromatic roots of graphs of order (that is, with
vertices), relatively little is known about the maximum imaginary part of such
graphs. We prove that the maximum imaginary part can grow linearly in the order
of the graph. We also show that for any fixed , almost every
random graph in the Erd\"os-R\'enyi model has a non-real root.Comment: 4 figure
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