Further results on sum index and difference index of graphs

Let G be a simple graph with at least one vertex. Assigning distinct integers to the vertices of G leads to two natural ways of producing labels on its edges, namely through the sum and the absolute difference of the labels of adjacent vertices. The range sizes of these induced edge values vary with the chosen vertex assignment. The sum index and difference index of a graph are introduced as the smallest attainable sizes of these ranges over all possible injective vertex labelings. This study establishes exact values of these indices for several prominent families of graphs, including cocktail party graphs, barbell graphs, pineapple graphs, generalized pineapple graphs, generalized fan graphs, and the graph class NC(Fm,n). The influence of certain graph operations on these indices is also examined. In addition, the sum index of the generalized Petersen graph P(n,3) is derived.