The edge cover polynomials of cubic graphs of order 10

Abstract

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 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. Also we construct infinite families of graphs whose edge cover polynomials have only roots $-1$ and $0$.