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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 oneedge 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 ofedge coverings of $G$ of size $i$, and $\rho(G)$ 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 Petersengraph) are determined uniquely by their edge cover polynomials. Also weconstruct infinite families of graphs whose edge cover polynomials have onlyroots $-1$ and $0$.

The edge cover polynomials of cubic graphs of order 10