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) ∪ E(G) → {?1, 1}. The weight of f is w(f) = Σ x∈V(G)∪E(G) f(x). For an element x ∈ V (G) ∪ E(G), we define $f[x] = \sum\nolimits_{y \in N_T [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 number γ 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 of γ s * (G) when G is C n and P n .

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.