### THE MINIMUM HARMONIC INDEX FOR BICYCLIC GRAPHS WITH GIVEN DIAMETER

#### 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.

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