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On the Domination Polynomials of Friendship Graphs

Let $G$ be a simple graph of order $n$. The {\em domination polynomial} of $G$ is the polynomial ${D(G, x)=\sum_{i=0}^{n} d(G,i) x^{i}}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Let $n$ be any positive integer and $F_n$ be the Friendship graph with $2n + 1$ vertices and $3n$ edges, formed by the join of $K_{1}$ with $nK_{2}$. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every $n\geq 2$, $F_n$ is not $\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real domination roots are only $-2$ and $0$. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the $n$-book graphs $B_n$, formed by joining $n$ copies of the cycle graph $C_4$ with a common edge.