Geodesic
On a Conjecture of Harmonic Index and Diameter of Graphs
Kragujevac Journal of Mathematics, Tome 40 (2016) no. 1, p. 73

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The Harmonic index $H(G)$ of a graph $G$ is defined as the sum of the weights $\dfrac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of the vertex $u$ in $G$. In this work, we prove the conjecture $H(G)-D(G) \geq \dfrac{5}{6}-\dfrac{n}{2}$ given by Liu in 2013 when G is a unicyclic graph by giving a better bound, namely, $H(G)-D(G)\geq \dfrac{5}{3}-\dfrac{n}{2}$.
Classification : 05C07 05C12
Keywords: Harmonic index, diameter, unicyclic graph. 
J. Amalorpava Jerline; L. Benedict Michaelraj. On a Conjecture of Harmonic Index and Diameter of Graphs. Kragujevac Journal of Mathematics, Tome 40 (2016) no. 1, p. 73 . http://geodesic.mathdoc.fr/item/KJM_2016_40_1_a5/
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