The minimum harmonic index for bicyclic graphs with given diameter

TL;DR

This paper determines the lowest possible harmonic index for bicyclic graphs with a specified number of vertices and diameter, and characterizes all graphs that achieve this minimum.

Contribution

It introduces the exact minimum harmonic index for bicyclic graphs with given order and diameter, and characterizes the extremal graphs.

Findings

Identified the minimum harmonic index for bicyclic graphs of given order and diameter.

Characterized all bicyclic graphs that attain this minimum harmonic index.

Abstract

The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.