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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.\n  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 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. 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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.\n  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 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. 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