On the bivariate permanent polynomials of graphs

In 1980, Balasubramanian and Parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this polynomial is a graph characterizing polynomial, that is, any two graphs with the same bivariate permanent polynomial are isomorphic. In this paper, we give counterexamples to this conjecture by a computer search. Furthermore, we show that several well-known families of graphs are determined by the bivariate permanent polynomial: complete graphs, complete bipartite graphs, regular complete multipartite graphs, cycles and their complements.