# On the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges

## TL;DR

This paper investigates the extremal values of the first and second multiplicative Zagreb indices in graphs with a fixed number of cut edges, providing characterizations and extending known results.

## Contribution

It characterizes graphs with maximum and minimum Zagreb indices among those with specified vertices and cut edges, extending previous findings.

## Key findings

- Identified graphs with extremal Zagreb indices for given cut edges.
- Provided formulas and characterizations for maximum and minimum indices.
- Extended known results in graph theory related to Zagreb indices.

## Abstract

For a molecular graph, the first multiplicative Zagreb index $\Pi_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $\Pi_2$ is equal to the product of the endvertex degree of each edge over all edges. Denote by $\mathbb{G}_{n,k}$ the set of graphs with $n$ vertices and $k$ cut edges. In this paper, we explore graphs in terms of a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges are provided. Furthermore, we characterize graphs with the largest and smallest $\Pi_1(G)$ and $\Pi_2(G)$ in $\mathbb{G}_{n,k}$, and our results extend and enrich some known conclusions.