TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS

Let G = (V, E) be a simple graph of order n. The total dominating set 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 γt(G). The total domination polynomial of G is the polynomial Dt(G, x) = ∑ dt(G, i)xi, where dt(G, i) is the number of total dominating sets of G having size i. A root of Dt(G, x) is called a total domination root of G. Let G be a connected graph constructed from pairwise disjoint connected graphs G1, . . . , Gk by selecting a vertex of G1, a vertex of G2, and identify these two vertices. Then continue in this manner inductively. We say that G is obtained by point-attaching from G1, . . . , Gk and that Gi’s are the primary subgraphs of G. In this paper, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials.