完全图全符号控制数的较小上界和下确界

The Total Signed Domination Number Smaller Upper and Greatest Lower Bounds of Complete Graph

设图G=G(V,E),令函数f∶V∪E→{-1,1},f的权w(f)=∑x∈V∪Ef[x],对V∪E中任一元素,定义f[x]=∑y∈NT[x]f(y),这里NT[x]表示V∪E中x及其关联边、邻点的集合.图G的全符号控制函数为f∶V∪E→{-1,1},满足对所有的x∈V∪E有f[x]1,图G的全符号控制数γT(G)就是图G上全符号控制数的最小权,称其f为图G的γT-函数.本文得到了完全图全符号控制数的一个较小上界和下确界.

Let G=G（V,E）be a graph.For a function f：V∪E→{-1,1},the weight of f is w（f）=∑（x∈V∪E）f[x].For an element x in V∪E ,we define f{x}=∑（y∈T{x}）f（y）,where NT{x} denote the set of x and the adjacent and incident elements of x∈V∪E.A total signed dominating function of G is a function f：V∪E→{-1,1},such that f[x]≥1 for all x∈V∪E.The total signed domination number γT（G） of G is the minimum weight of a total signed dominating function on G.The total signed dominating function of a weight γT（G） we call γT（G）-function of G.In this paper the smaller upper and greatest lower bounds of the total signed domination number of com plete Graph are obtained.