On the roots of total domination polynomial of graphs

TL;DR

This paper investigates the roots of the total domination polynomial of graphs, establishing bounds on their location in the complex plane and characterizing integer roots for graphs with high minimum degree.

Contribution

It provides new bounds on the roots of the total domination polynomial and characterizes integer roots for graphs with minimum degree at least two-thirds of the order.

Findings

All roots lie within a specific circle in the complex plane.

Integer roots are limited to a small set for graphs with high minimum degree.

The results connect graph properties with polynomial root locations.

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 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 $\{-3,-2,-1,0\}$.