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

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Omega provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Omega is defined as the product of the sum of the ath powers of the vertex degrees of Omega, where alpha is a real number such that alpha (sic) 0 and alpha (sic) 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 Pi(alpha)(1) for predicting the pi-electronic energy E-pi(beta) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of alpha for which the predictive potential of Pi(alpha)(1) with E-pi for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Pi(alpha)(1) correlates with E-pi of lower benzenoid hydrocarbons with correlation coefficient rho = -0.998, if alpha = 0.00496. In QSPR modeling, the value rho = -0.998 is considered to be considerably significant.