On Total Domination Polynomials of Certain Graphs

Abstract

We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined by, where d t (G, i) is 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 anyAnd, for any k-regular graph of order n,We have calculated the total domination polynomial of Petersen graph D t (P, x) = 10x 4 + 72x 5 + 140x 6 + 110x 7 + 45x 8 + 10x 9 + x 10 .Also, for any two vertices u and v of a k-regular graph H with N(u) ≠ N(v) and if D t (G, x) = D t ( H, x ), then G is also a k-regular graph.