Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling

Abstract

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Ω is defined as the product of the sum of the αth powers of the vertex degrees of Ω, where α is a real number such that α≠0 and α≠1. The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index Π1α for predicting the π-electronic energy Eπ(β) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π1α with Eπ for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π1α correlates with Eπ of lower benzenoid hydrocarbons with correlation coefficient ρ=−0.998, if α=−0.00496. In QSPR modeling, the value ρ=−0.998 is considered to be considerably significant.

Keywords

• multiplicative Zagreb index
• graph
• unicyclic graph
• bicyclic graph
• extremal values