The multiplicative Zagreb indices of graphs with given connectivity or number of pendant vertices

TL;DR

This paper investigates the extremal properties of multiplicative Zagreb indices in graphs with specified connectivity and pendant vertices, providing characterizations and bounds that extend previous results.

Contribution

It characterizes extremal graphs for multiplicative Zagreb indices under connectivity and pendant vertex constraints, offering new bounds and extending known conclusions.

Findings

Identifies extremal graphs for given connectivity and pendant vertices.

Provides maximum and minimum values of the indices.

Extends and enriches existing theoretical results.

Abstract

For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal $\Pi_{1}(G)$ and $\Pi_{2}(G)$ in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most $k$ and $p$ pendant vertices. In addition, the maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ are provided. Our results extend and enrich some known conclusions.