### On the Number of Perfect Matchings of Generalized Theta Graphs and the Edge Cover Polynomials of Friendship Graphs

#### Abstract

An edge covering of a graph is a set of edges such that every vertex of

the graph is incident to at least one edge of the set. Let $G$ be a simple graph with $m$ edges. The edge

cover polynomial of $G$ is the polynomial $E(G,x)=\sum_{i=1}^{m}

e(G,i) x^{i}$, where $e(G,i)$ is the number of edge coverings of $G$ of size

$i$. Let $t$ be a positive integer and $F_t$

be the friendship (or Dutch windmill) graph with $2t+1$

vertices and $3t$ edges. In this paper we study the edge cover polynomial of friendship graphs. We show that the friendship graphs are determined by their edge cover polynomials. We find that all non-zero roots of the edge cover polynomial of friendship graphs are simple. Finally we prove that the edge coverĀ polynomials of friendship graphs are unimodal.

the graph is incident to at least one edge of the set. Let $G$ be a simple graph with $m$ edges. The edge

cover polynomial of $G$ is the polynomial $E(G,x)=\sum_{i=1}^{m}

e(G,i) x^{i}$, where $e(G,i)$ is the number of edge coverings of $G$ of size

$i$. Let $t$ be a positive integer and $F_t$

be the friendship (or Dutch windmill) graph with $2t+1$

vertices and $3t$ edges. In this paper we study the edge cover polynomial of friendship graphs. We show that the friendship graphs are determined by their edge cover polynomials. We find that all non-zero roots of the edge cover polynomial of friendship graphs are simple. Finally we prove that the edge coverĀ polynomials of friendship graphs are unimodal.

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