Abstract:We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined byis the cardinality of total dominating sets of G of size i, and γ t (G) is the total domination number of G.In [7] We have obtained some properties of D t (G, x) and its coefficients. Also, we have calculated the total domination polynomials of complete graph, complete bipartite graph, join of two graphs and a graph consisting of disjoint components. In this paper, we presented for any two isomorphic graphs the total domination polynomials are same, but the converse is not true. Also, we proved that for any vertex transitive graph of order n and for then G is also a k-regular graph.

Let and be two simple graphs. Let (,  1) and (,  2) be two vertex measure spaces. In this paper we introduce a σ algebra  1  2 , which consists of all vertex induced sub graphs of , and it contains every vertex measurable rectangle graph of the form H 1 H 2 , H 1 ∈  1 and H 2 ∈  2. Here, we prove  1  2 is the smallest σ algebra of such that the maps and defined by and for all vertex measurable graphs H in and K in respectively are measurable.