On the roots of total domination polynomial of graphs

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of total domination polynomial of some graphs. We show that all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and the radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence we prove that if $\delta\geq \frac{2n}{3}$, then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.