A Graph Polynomial for Independent Sets of Bipartite Graphs

Abstract

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.