Bounds on some monotonic topological indices of bipartite graphs with a given number of cut edges

TL;DR

This paper establishes bounds on various monotonic topological indices of bipartite graphs based on the number of cut edges, identifying extremal graphs and analyzing how these indices change with edge addition.

Contribution

It provides new bounds and characterizations for monotonic topological indices in bipartite graphs considering cut edges, extending understanding of their extremal properties.

Findings

Derived bounds for Wiener, hyper-Wiener, and Harary indices.

Characterized extremal bipartite graphs with given cut edges.

Analyzed monotonic behavior of topological indices with edge addition.

Abstract

Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\not\in G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this article, we present lower or upper bounds for some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum of bipartite graphs in terms of the number of cut edges, and characterize the corresponding extremal graphs, respectively.