On the roots of total domination polynomial of graphs. (18th May 2020) - Jafari, Nasrin; Alikhani, Saeid
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.