On the sharp upper and lower bounds of multiplicative Zagreb indices of graphs with connectivity at most k

TL;DR

This paper investigates the maximum and minimum bounds of multiplicative Zagreb indices for graphs with limited connectivity, characterizing extremal graphs and extending existing results in graph theory.

Contribution

It provides sharp bounds and characterizations for multiplicative Zagreb indices of graphs with connectivity at most k, extending previous findings.

Findings

Identified maximum and minimum values of $ ext{prod}_1(G)$ and $ ext{prod}_2(G)$ for graphs with connectivity ≤ k.

Characterized extremal graphs achieving these bounds.

Extended known results in the study of Zagreb indices and graph connectivity.

Abstract

For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G) $ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of graphs with connectivity at most $k$ are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions.