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

Mohammad Reza Oboudi


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.

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