### Domination polynomial of generalized book graphs

#### Abstract

Let $G$ be a simple graph of order $n$.

The 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 $B_n$ be the {\em $n$-book graphs}, formed by joining $n$ copies of the cycle graph $C_4$ with a common edge. In this paper, we study the domination polynomials of some generalized book graphs. In particular we examine the domination roots of these families, and find the limiting curve for the roots.

The 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 $B_n$ be the {\em $n$-book graphs}, formed by joining $n$ copies of the cycle graph $C_4$ with a common edge. In this paper, we study the domination polynomials of some generalized book graphs. In particular we examine the domination roots of these families, and find the limiting curve for the roots.

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