The cover polynomial introduced by Chung and Graham is a twovariate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is #...

Interpretations for evaluations of the Tutte polynomial T (G;x, y) of a graph G are given at a number of points on the hyperbolae Hq = {(x, y) | (x − 1)(y − 1) = q}, for q a positive integer — points at which there are usually no other similarly meaningful graphical interpretations. Further, when q is a prime power, an alternative interpretation for the evaluation of the Tutte polynomial at (1−...

For a quiver $Q$ with underlying graph $\Gamma$ , we take $ {\mathcal {M}}$ an associated toric Nakajima variety. In this article, give direct relation between specialization of the Tutte polynomial Kac and Poincaré . We do by giving cell decomposition indexed spanning trees ‘geometrizing’ deletion contraction operators on graphs. These relations have been previously established in Hausel–Sturm...

We associate two modules, the \(G\)-parking critical module and toppling module, to an undirected connected graph \(G\). The are canonical modules (with suitable twists) of quotient rings well-studied function ideal ideal, respectively. For each we establish a Tutte-like short exact sequence relating associated \(G\), edge contraction \(G/e\) deletion \(G \setminus e\) (\(e\) is non-bridge). ob...

We discuss reducing the number of steps involved in computing the Tutte polynomial of a matroid by using series and parallel reductions in conjunction with the usual deletion and contraction operations. c © 2000 Elsevier Science B.V. All rights reserved.