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

Sakander Hayat () and Farwa Asmat
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Sakander Hayat: Faculty of Science, Universiti Brunei Darussalam, Jln Tungku Link, Gadong BE1410, Brunei
Farwa Asmat: School of Mathematical Sciences, Peking University, Beijing 100871, China

Mathematics, 2023, vol. 11, issue 10, 1-14

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 α th 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 Π 1 α for predicting the π -electronic energy E π ( β ) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π 1 α with E π for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π 1 α correlates with E π 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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