Black-white polynomials of graphs and generating functions

Abstract

Let G be a graph. The black-white polynomial W_G(t) enumerates colorings of the vertices of G with two colors (black and white), where the power of t keeps track of how many white vertices have an even number of black neighbors. Such polynomials appear in quantum information theory, where they are used to capture properties of the entanglement in certain quantum states described by graphs. In this paper we describe how to use generating functions to compute these polynomials for various families X of graphs. Our main results are the following: (i) we describe some constructions under which X leads to a rational generating function; (ii) we use a matrix model to construct the exponential generating function of the black-white polynomials of all graphs; and (iii) we generalize a construction of Wright to build exponential generating functions of black-white polynomials for graphs of a given loop number.