On some bounds of first Gourava index for Ψ-sum graphs

Graph operations play a significant role in constructing new and valuable graphs and capturing intermolecular forces between atoms and bonds of a molecule. In mathematical chemistry and chemical graph theory, a topological invariant is a numeric value extracted from the molecular graph of a chemical compound using a mathematical formula involving vertex degrees, distance, spectrum, and their combination. An intriguing problem in chemical graph theory is figuring out the lower and the upper bound on pertinent topological indices among a particular family of graphs. The first Gourava index for a graph [Formula: see text] is denoted and defined as [Formula: see text] Recently, Kulli studied and derived formulas of the first Gourava index for four graph operations. We proved with the help of counter-examples that the results provided by Kulli produce inaccurate values when compared with exact values. In this paper, we determined the exact formulas and bounds of the first Gourava index for [Formula: see text]-sum graphs. Besides, we presented diverse examples to support our results.