The sum numbers and the integral sum numbers of the graph Kn+1∖E(K1,r)Kn+1∖E(K1,r)

Let NN denote the set of all positive integers. The sum graph G+(S)G+(S) of a finite subset S⊂NS⊂N is the graph (S,E)(S,E) with uv∈Euv∈E if and only if u+v∈Su+v∈S. A simple graph GG is said to be a sum graph if it is isomorphic to a sum graph of some S⊂NS⊂N. The sum number σ(G)σ(G) of GG is the smallest number of isolated vertices which when added to GG result in a sum graph. Let ZZ denote the set of all integers. The integral sum graph G+(S)G+(S) of a finite subset S⊂ZS⊂Z is the graph (S,E)(S,E) with uv∈Euv∈E if and only if u+v∈Su+v∈S. A simple graph GG is said to be an integral sum graph if it is isomorphic to an integral sum graph of some S⊂ZS⊂Z. The sum number ζ(G)ζ(G) of GG is the smallest number of isolated vertices which when added to GG result in an integral sum graph. In this paper, we prove that σ(Kn+1∖E(K1,r))=ζ(Kn+1∖E(K1,r))={2n−2,r=1,2n−3,2≤r≤n−1,2n−4,r=n.