A family of sparse graphs of large sum number

Given an integer r ⩾ 0, let Gr, = (Vr, E) denote a graph consisting of a simple finite undirected graph G = (V, E) of order n and size m together with r isolated vertices Kr. Then | V | = n, |Vr| = n+r, and |E| = m. Let L:Vr → Z+ denote a labelling of the vertices of Gr with distinct positive integers. Then Gr is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of Vr, (u, v) ϵE if and only if …