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

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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. 
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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/