New Results for the Martin Polynomial

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a universal skein-type graph polynomial P(G) which is a Hopf map, and then using the recursion and induction which naturally arise from the Hopf algebra structure to extend known properties. Specifically, when P(G) is evaluated by substituting s for all cycles and 0 for all tails, then P(G) equals sM(G; s+2) for all Eulerian graphs G. The Hopf-algebraic properties of P(G) are then used to extract new properties of the Martin polynomial, including an immediate proof for the formula for M(G; s) on disjoint unions of graphs, combinatorial interpretations for M(G; 2+2) and M(G; 2&2) with k # Z , and a new formula for the number of Eulerian orientations of a graph in terms of the vertex degrees of its Eulerian subgraphs. 1998