Exponential Zagreb indices and operations on graphs

The concept of the exponential of vertex-degree-based invariants was put forward by Rada [Exponential vertex-degree-based topological indices and discrimination, MATCH Commun. Math. Comput. Chem. 82(1) (2019) 29–41] with the purpose of investigating the discriminating ability of this category of topological indices. In view of this, the exponential of the two well-known Zagreb indices are formulated for a simple connected graph [Formula: see text] as [Formula: see text] where [Formula: see text] and [Formula: see text] stand for the vertex set and edge set of [Formula: see text], respectively, and [Formula: see text] indicates the degree of the vertex [Formula: see text] in [Formula: see text]. In this paper, we give sharp upper and lower bounds on the exponential Zagreb indices of several families of graph operations such as the sum, Indu–Bala product, vertex-corona product, edge-corona product, Cartesian product, strong product, and lexicographic product in terms of the exponential Zagreb indices and some graph parameters of their factors. The necessary and sufficient conditions for the inequalities to be transformed into equality are also examined. In the special case that both factors are regular graphs, exact formulas for the exponential Zagreb indices of all aforementioned operations are provided. The obtained results are illustrated by several interesting examples from certain special families of graphs.