Extremal c-Cyclic Graphs with Respect to the General Multiplicative First Zagreb Index

Abstract

The general multiplicative first Zagreb index of a simple graph $H$ is
expressed as the product of the weights $(\deg_H(\omega))^\alpha$ over all vertices $\omega$ of $H$, where $\deg_H(\omega)$ shows the degree of $\omega$, and $\alpha \neq 0$ is a real number. The cyclomatic number of a connected graph $H$ is given by $c=\epsilon-\nu+1$, where $\epsilon$ and $\nu$ are the size and order of $H$, respectively. In this paper, we present sharp bounds for the general multiplicative first Zagreb index of simple connected graphs with cyclomatic number $c$ focusing on the cases when $c$=0, 1, and 2. We also extend our findings to molecular trees and to all simple connected graphs with the maximum degree $\Delta$ and cyclomatic number $c$, where $\Delta\geq 2c$. In addition, we identify the graphs reaching these bounds.