The sum number and the integral sum number of a generalization of the dancing version of the cocktail party graph

Abstract

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 uv ⋯ E if and only if u + v ⋯ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S⊂N. The sum number σ(G) of G is the smallest number of isolated vertices, which result in a sum graph when added to G. Let Z denote the set of all integers. F. Harary introduced the notions of the integral sum graph and the integral sum number of a graph respectively by extending N to Z in the definitions of the sum graph and the sum number. In this paper, we investigate and determine the sum number and the integral sum number of a generalization of the dancing version of the cocktail party graph, i.e., K_n,n\E(rK₂), where n ≥ 6 and 1 ≤ r ≤ n.