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 $Y_M(q,t)$ of the Tutte polynomial of a matroid $M$ 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 $M$. We show that the coefficients of $Y_M(1-p,t)$ 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 $M$. 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 $n$ (that is, with $n$ 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 $p \in (0,1)$, almost every random graph $G$ in the Erd\"os-R\'enyi model has a non-real root.Comment: 4 figure