Geodesic
The Harmonic Index of Unicyclic Graphs with Given Matching Number
Kragujevac Journal of Mathematics, Tome 38 (2014) no. 1, p. 173

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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 $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. In this paper, we determine the graph with minimum harmonic index among all unicyclic graphs with a perfect matching. Moreover, the graph with minimum harmonic index among all unicyclic graphs with a given matching number is also determined.
Classification : 05C12 92E10
Keywords: Harmonic index, Matching number, Unicyclic graph 
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Jian-bo Lv; Jianxi Li; Wai Chee Shiu. The Harmonic Index of Unicyclic Graphs with Given Matching Number. Kragujevac Journal of Mathematics, Tome 38 (2014) no. 1, p. 173 . http://geodesic.mathdoc.fr/item/KJM_2014_38_1_a12/