Abstract :
Given a graph G=(V,E), if e=uv∈E, then the closed edge-neighbourhood of e is denoted by N[e]={u′v′∈E|u′=u or v′=v}. A function f : E→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑e′∈N[e]f(e′)⩾1 for every e∈E. The signed edge domination number γs′(G) of G is defined as γs′(G)=min{∑e∈E f(e) | f is an SEDF of G}. Let Ψ(m)=min{γs′(H)|H is a graph with |E(H)|=m}. In this paper, we determine the exact value of Ψ(m) for each positive integer m. That is:Ψ(m)=21324m+25+6m+56−m,where ⌈x⌉ denotes the ceiling of x. In addition, we also characterize all connected graphs G with γs′(G)=|E(G)|.
