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 analyze 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. We propose a natural Markov chain to approximately evaluate the polynomial for a range of parameters. We prove an upper bound on the mixing time of the Markov chain on trees. As a by-product we show that the ``single bond flip'' Markov chain for the random cluster model is rapidly mixing on constant tree-width graphs.
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