Estimating the First, Hyper-Zagreb Index for Direct Product of F-Sum Graphs

A chemical graph is an illustration of a chemical compound’s structural formula, where the edges indicate covalent bonding and the vertices depict atoms. In the study of (molecular) graphs, Zagreb indices and their variations are the most studied and well-investigated invariants due to their chemical applicability. To probe and investigate the most promising lower and upper bound of the appropriate topological index for a graph family is a problem of primal significance in extremal graph theory. Deng et al. computed the closed-form formulas for the Zagreb indices M1 and M2 for four novel operations known as F-sum graphs, proposed by Eliasi and Taeri in “Discrete Appl. Math. (2009) 157.” Subsequently, Buragohain et al. derived the formulas for M1 and M2 of four sum graphs in the context of the direct product. In this article, we investigate F-sum graphs formed through four newly defined graph operations associated with the direct product and establish lower bounds and upper bounds on the first hyper-Zagreb index HM1 for these graphs. Finally, we validate our derived bounds by applying them to various examples, demonstrating the correctness and applicability of the results.