On a Conjecture of Harmonic Index and Diameter of Graphs

J. Amalorpava Jerline, L. Benedict Michaelraj

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, $H(G)-D(G)\geq \dfrac{5}{3}-\dfrac{n}{2}$.