On the Roots of Total Domination Polynomial of Graphs, Ii

Abstract

Let $G = (V, E)$ be a simple graph of order $n$. A total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$ is minimum cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the 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 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 $\{-3,-2,-1,0\}$.

Some new results on the total domination polynomial of a graph

“…Proof. Let G be a 3-connected graph and D t (G, x) = x γt (x + 2) β , where γ t + β = n. So by Theorem 4.1, we have d t (G, n − 1) = n = 2β, and so 2 , and so G = K 4 , GP . On the other hand, for every k = 2, .…”

“…Also the first authors has studied graphs with few domination roots in [1]. In [2] we have shown that all roots of D t (G, x) lie in the circle with center (−1, 0) and the radius δ √ 2 n − 1, where δ is the minimum degree of G. Also we proved that for a graph G of order n, if δ ≥ 2n 3 , then every integer root of D t (G, x) lies in the set {−3, −2, −1, 0}. Motivated by these integer roots, and a conjecture in [2] which states that for every integer root r of D t (G, x), r ∈ {−3, −2, −1, 0}, we study graphs with exactly two total domination roots {−1, 0}, {−2, 0} and {−3, 0}, in this section.…”

“…We refer the reader to [14] and its references for more information in roots of graph polynomials. In [2] we have shown that all roots of D t (G, x) lie in the circle with center (−1, 0) and the radius δ √ 2 n − 1, where δ is the minimum degree of G. Also we proved that for a graph G of order n, if δ ≥ 2n 3 , then every integer root of D t (G, x) lies in the set {−3, −2, −1, 0}. As usual we denote the complete graph, path and cycle of order n by K n , P n and C n , respectively.…”

On the independent domination polynomial of a graph

“…For many graph polynomials, their roots have attracted considerable attention, both for their own sake, as well for what the nature and location of the roots imply. The roots of the chromatic polynomial, independence polynomial, domination polynomial and total domination polynomials have been studied extensively [1,2,5,8,10,11,12,14,26]. We investigate here independence domination roots, that is, the roots of independent domination polynomials.…”

Connected coalitions in graphs

“…The study of coalition graphs, particularly for paths, cycles, and trees, was conducted in [9]. The concept of total coalition was introduced and explored in [1], while the coalition parameter for cubic graphs of order at most 10 was investigated in [2].…”