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 \gamma_t(G). The total domination polynomial of G is the polynomial D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i, where d_t(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 \mathcal{D}_t-equivalent, if D_t(G,x)=D_t(H,x). The equivalence class of G, denoted [G], is the set of all graphs \mathcal{D}_t-equivalent to G. A polynomial \sum_{k=0}^n a_kx^k is called unimodal if the sequence of its coefficients is unimodal, that means there is some k \in \{0, 1, \ldots , n\}, such that a_0 \leq \ldots \leq a_{k-1} \leq a_k\geq a_{k+1} \geq \ldots \geq a_n. In this paper, we investigate \mathcal{D}_t-equivalence classes of some graphs. Also, we introduce some families of graphs whose total domination polynomials are unimodal. The \mathcal{D}_t-equivalence classes of graphs of order \leq 6 are presented in the appendix.

MORE ON TOTAL DOMINATION POLYNOMIAL AND Dt-EQUIVALENCE CLASSES OF SOME GRAPHS Keywords:

total domination polynomial , equivalence class , unimodal