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 (Formula presented.) provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph (Formula presented.) is defined as the product of the sum of the (Formula presented.) th powers of the vertex degrees of (Formula presented.), where (Formula presented.) is a real number such that (Formula presented.) and (Formula presented.). 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 (Formula presented.) for predicting the (Formula presented.) -electronic energy (Formula presented.) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of (Formula presented.) for which the predictive potential of (Formula presented.) with (Formula presented.) for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that (Formula presented.) correlates with (Formula presented.) of lower benzenoid hydrocarbons with correlation coefficient (Formula presented.), if (Formula presented.). In QSPR modeling, the value (Formula presented.) is considered to be considerably significant.