Shahrood University of TechnologyJournal of Algebraic Systems2345-51282220150201ON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS9710835910.22044/jas.2015.359ENS.AlikhaniDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran.S.JahariDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran.Journal Article20140430Let $G$ 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<br />$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$, where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and<br />$rho(G)$ 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.http://jas.shahroodut.ac.ir/article_359_03bd853b0f975a60d986af404d928abd.pdf