Tutte polynomials for regular oriented matroids

Abstract

The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid N, we associate a polynomial invariant AN(q,y,z), which we call the A-polynomial. The A-polynomial has the following interesting properties among many others:•a specialization of AN gives the Tutte polynomial of the underlying unoriented matroid N_,•when the oriented matroid N corresponds to an unoriented matroid (that is, when the elements of the ground set come in pairs with opposite orientations), the invariant AN is equivalent to the Tutte polynomial of this unoriented matroid (up to a change of variables),•the invariant AN detects, among other things, whether N is acyclic and whether N is totally cyclic. We explore various properties and specializations of the A-polynomial. We show that some of the known properties of the Tutte polynomial of matroids can be extended to the A-polynomial of regular oriented matroids. For instance, we show that a specialization of AN counts all the acyclic orientations obtained by reorienting some elements of N, according to the number of reoriented elements.Let us mention that in a previous article we had defined an invariant of oriented graphs that we called the B-polynomial, which is also a generalization of the Tutte polynomial. However, the B-polynomial of an oriented graph N is not equivalent to its A-polynomial, and the B-polynomial cannot be extended to an invariant of regular oriented matroids.