This talk will cover the Martin Polynomial and some of its properties. Recall that an Eulerian graph can be covered by a cycle which traverses each of its edges exactly once. The Martin polynomial of a graph is a one variable polynomial whose coefficients ennumerate the families of Eulerian subgraphs of the original graph. However, more information about the graph is encoded in the polynomial, but, as with many other graph polynomials, extracting it can be very difficult. A new identity for the Martin polynomial makes it possible to get combinatorial interpretations for valuations of the Martin polynomial fairly easily by using induction. The talk will conclude with a brief description of how the new results for the Martin polynomial derive from its being a special case of a much more general polynomial with especially rich algebraic properties.