完全图全符号控制数的较小上界和下确界

设图G=G(V,E),令函数f∶V∪E→{-1,1},f的权w(f)=∑x∈V∪Efx],对V∪E中任一元素,定义fx]=∑y∈NTx]f(y),这里NTx]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fx]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.

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

Let G=G(V, E)be a graph. For a function f∶V∪E→{-1, 1},the weight of fis w(f)=∑x∈V∪Efx].For an element xin V∪E,we define fx]=∑y∈N_Tx]f(y), where N_Tx]denote the set of xand the adjacent and incident elements of x∈V∪E.A total signed dominating function of Gis a function f∶V∪E→{-1, 1},such that fx]1for all x∈V∪E.The total signed domination number γ_T(G)of Gis the minimum weight of a total signed dominating function on G.The total signed dominating function of a weight γ_T(G)we call γ_T(G)-function of G.In this paper the smaller upper and greatest lower bounds of the total signed domination number of Complete Graphare obtained.