The sum numbers and the integral sum numbers of the graph

Let 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 u v ∈ 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 u v ∈ 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 ( K n + 1 ∖ E ( K 1 , r ) ) = ζ ( K n + 1 ∖ E ( K 1 , r ) ) = { 2 n − 2 , r = 1 , 2 n − 3 , 2 ≤ r ≤ n − 1 , 2 n − 4 , r = n .