The sum numbers and the integral sum numbers of the graph K n\E(Pn-1)

Abstract

Let N denote the set of all positive integers. The sum graph G ⁺(S) of a finite subset S C N is the graph (S, E) with uv 6 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 C 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 C 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 ((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 determine that ξ(K _n\E(P_n-1))=0, n=4,5,6 2n-7, n≥7 and σ(K _n\E(P_n-1))=1, n=4 3, n=5 5, n=6 2n-7, n≥7.