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)=∑i=γₜ(G)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 ∑ₖ₌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 ≤ … ≤ 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.