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）∪E（G）→{-1,1}.The weight of f is W（f）=∑x∈V（G）∪E（G）)f（X）.For an element x∈V（G）∪E（G）,we define f[x]=∑y∈NT[x]f（y）.A total signed domination function of G is a function f:V（G）∪E（G）→{-1,1} such that f[x]≥1 for all x∈V（G）∪E（G）.The total signed domination numberγs*（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 ofγs*（G）when G is Cn and Pn.
Xin-zhong LU Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China
