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Article

on the edge cover polynomial of certain graphs

saeid alikhani, sommayeh jahari

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 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.

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