On the roots of total domination polynomial of graphs

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 γt(G). The total domination polynomial of G is the polynomial , where dt (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 Dt (G, x) lie in the circle with center (–1, 0) and radius , where d is the minimum degree of G. As a consequence, we prove that if , then every integer root of D t (G, x) lies in the set {–3, –2, –1, 0}.