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$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we study theedge cover polynomials of cubic graphs of order $10$.We show that all cubic graphs of order $10$ (especially the Petersen graph) aredetermined uniquely by their edge cover polynomials.