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

AbstractLet N denote the set of all positive integers. The sum graph G+(S) of a finite subset S⊂N is the graph (S,E) with uv∈E if and only if u+v∈S. A simple graph G is said to be a sum graph if it is isomorphic to a sum graph of some S⊂N. The sum number σ(G) of G is the smallest number of isolated vertices which when added to G result in a sum graph. Let Z denote the set of all integers. The integral sum graph G+(S) of a finite subset S⊂Z is the graph (S,E) with uv∈E if and only if u+v∈S. A simple graph G is said to be an integral sum graph if it is isomorphic to an integral sum graph of some S⊂Z. The sum number ζ(G) of G is the smallest number of isolated vertices which when added to G 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