On the roots of total domination polynomial of graphs, II

Abstract

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 is denoted by ${\gamma }_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)={\sum }_{i={\gamma }_t(G)}^n{d}_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. The set of total domination roots of graph $G$ is denoted by $Z(D_t(G,x))$. In this paper we show that $D_t(G,x)$ has $\delta -2$ non-real roots and if all roots of $D_t(G,x)$ are real then $\delta \le 2$, where $\delta$ is the minimum degree of vertices of $G$. Also we show that if $\delta \ge 3$ and $D_t(G,x)$ has exactly three distinct roots, then $Z(D_t(G,x))\subseteq \{0,-2\pm \sqrt{2}i,\frac{-3\pm \sqrt{3}i}{2}\}$. Finally we study the location roots of total domination polynomial of some families of graphs.

10 pages, 5 figures