A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G) and an edge set E(G). A total signed domination function of G is a function f : V(G) U E(G) -> [-1,1}. The weight of f is w(f) = Sigma(x epsilon V(G)UE(G)) f (x). For an element x E V(G) U E(G), we define f [x] = E,,,,[x] f (y). A total signed domination function of G is a function f : V(G) U E(G), f -1, I} such that f [x] >= 1 for all x G V(G) U E(G). The total signed domination number gamma*(s)(G) of G is the minimum weight of a total signed domination function on G. In this paper, we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofy(s),* (G) when G is C. and P..