The (mod, integral) sum numbers of fans and Kn,n-E(nK2)Kn,n-E(nK2)

Let N(Z)N(Z) denote the set of all positive integers (integers). The sum graph G+(S)G+(S) of a finite subset S⊂N(Z)S⊂N(Z) is the graph (S,E)(S,E) with uv∈Euv∈E if and only if u+v∈Su+v∈S. A graph G   is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S⊂N(Z)S⊂N(Z). The (integral) sum number σ(G)(ζ(G))σ(G)(ζ(G)) of G is the smallest number of isolated vertices which when added to G   result in an (integral) sum graph. A mod sum graph is a sum graph with S⊂Zm⧹{0}S⊂Zm⧹{0} and all arithmetic performed modulo m   where m⩾|S|+1m⩾|S|+1. The mod sum number ρ(G)ρ(G) of G   is the least number ρρ of isolated vertices ρK1ρK1 such that G∪ρK1G∪ρK1 is a mod sum graph. In this paper, we prove that for n⩾3n⩾3, the n   spoked fan FnFn is an integral sum graph, ρ(F4)=1,ρ(Fn)=2ρ(F4)=1,ρ(Fn)=2 for n≠4n≠4, and σ(Fn)=2,n=4,3,n=3orn⩾6andneven,4,n⩾5andnodd.We also show that for Kn,n-E(nK2)(n⩾6)Kn,n-E(nK2)(n⩾6), ρ=n-2ρ=n-2, σ=2n-3σ=2n-3 and ζ=2n-5ζ=2n-5.