more on total domination polynomial and dt-equivalence classes of some graphs

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 some vertices of d. the total domination number of g is equal to the minimum cardinality of a total dominating set in g and is denoted by γt(g). the total domination polynomial of g is the polynomial dt(g, x) = ∑n i=γt(g) dt(g, i)x i , where dt(g, i) is the number of total dominating sets of g of size i. two graphs g and h are said to be total dominating equivalent or simply dt-equivalent, if dt(g, x) = dt(h, x). the equivalence class of g, denoted [g], is the set of all graphs dt-equivalent to g. a polynomial ∑n k=0 akx k is called unimodal, if the sequence of its coefficients is unimodal, that means there is some k ∈ {0, 1, . . . , n}, such that a0 ≤ . . . ≤ ak−1 ≤ ak ≥ ak+1 ≥ . . . ≥ an. in this paper, we investigate dt-equivalence classes of some graphs. also, we introduce some families of graphs whose total domination polynomials are unimodal. the dt equivalence classes of graphs of order ≤ 6 are presented in the appendix.