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 ₜ (G). The total domination polynomial of G is the polynomial Dₜ (G, x) =₈=䂻 (₆) ⁿ dₜ (G, i) xⁱ, 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 ₊=₀ⁿ aₖxᵏ is called unimodal if the sequence of its coefficients is unimodal, that means there is some k \0, 1, , n\, such that a₀ a₊-₁ aₖ a₊+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.