Some new results on the total domination polynomial of a graph

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Let G = (V, E) be a simple graph of order n. The total dominating set of G 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)=∑ᵢ₌ᵧₜ₍G₎ⁿ dₜ(G,i)xⁱ, where dₜ(G,i) is the number of total dominating sets of G of size i. A root of Dₜ(G,x) is called a total domination root of G. An irrelevant edge of Dₜ(G,x) is an edge e ∈ E, such that Dₜ(G, x) = Dₜ(G e, x). In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots {-3,0}, {-2,0} and {-1,0}.