ON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS

Let G be a simple graph of order n and size m. The edge covering of G is a set of edges such that every vertex of G is incident to at least one edge of the set. The edge cover polynomial of G is the polynomial E(G; x) = ?m i=(G) e(G; i)xi, where e(G; i) is the number of edge coverings of G of size i, and (G) is the edge covering number of G. In this paper we study the edge cover polynomials of cubic graphs of order 10. We show that all cubic graphs of order 10 (especially the Petersen graph) are determined uniquely by their edge cover polynomials.