Sharp bounds for the modified multiplicative Zagreb indices of graphs with connectivity at most k

Abstract

Zagreb indices and their modified versions of a molecular graph originates from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality.
Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects.

Let $\mathbb{V}_n^k$ (or $\mathbb{E}_n^k$, respectively) be a set of molecular graphs of $n$ vertices with vertex connectivity (or edge connectivity, respectively) at most $k$. In this paper, we explore
some properties of the modified first and second multiplicative Zagreb indices of graphs in $\mathbb{V}_n^k$ and $\mathbb{E}_n^k$.
By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in $\mathbb{V}_n^k$ and $\mathbb{E}_n^k$. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new
tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs.