NOTES ON THE HARMONIC INDEX OF GRAPHS

Abstract

For a connected graph $G$ of order $n$, the harmonic index of $G$ is the sum of weights

$$\frac{2}{d(u)+d(v)}$$

over all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. We prove that

$$H(G)\ge \frac{{\chi }_{DP}(G)-2}{2}+\frac{2({\chi }_{DP}(G)-1)}{n+{\chi }_{DP}(G)-2}+\frac{2(n-{\chi }_{DP}(G))}{n},$$

and this bound is sharp for all $n$ and $2\le {\chi }_{DP}(G)\le n$, where ${\chi }_{DP}(G)$ is the $DP$-chromatic number of $G$. This generalizes the previous lower bounds on $H(G)$. Moreover, we also determine the tree with minimum harmonic index among trees in ${\mathcal{𝒯}}_{n,l}$, where ${\mathcal{𝒯}}_{n,l}$ is the set of trees of order $n$ with a given segment sequence $l$.