On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices

TL;DR

This paper establishes the exact lower bounds for the first and second Zagreb indices of graphs with a specified number of cut vertices and at least one cycle, characterizing the extremal graphs.

Contribution

It solves an open problem by deriving sharp lower bounds for Zagreb indices in graphs with given cut vertices and characterizes the extremal graphs.

Findings

Sharp lower bounds for Zagreb indices are obtained.

Characterization of graphs with minimal Zagreb indices.

Results apply to graphs with specified cut vertices and cycles.

Abstract

The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about Zagreb indices of graphs with given number of cut vertices. The sharp lower bounds are obtained for these indices of graphs in $\mathbb{V}_{n,k}$, where $\mathbb{V}_{n, k}$ denotes the set of all $n$-vertex graphs with $k$ cut vertices and at least one cycle. As consequences, those graphs with the smallest Zagreb indices are characterized.