The Relation Between the Harmonic Index and Some Coloring Parameters

Abstract

Let H(G) be the harmonic index of a graph G, which is defined as:

$$\begin{aligned} H(G) = \sum _{uv \in E(G)}\frac{2}{d_{G}(u) + d_{G}(v)}. \end{aligned}$$

In this note, we define a new graph parameter $\xi (G)$ satisfying some properties and prove that $\xi (G) \le 2H(G)$, with equality if and only if G is a non-trivial complete graph, possibly plus some additional isolated vertices. In particular, $\xi (G)$ can be the chromatic number $\chi (G)$, the choice number $\chi _{\ell }(G)$, the DP-chromatic number $\chi _{\text {DP}}(G)$, the DP-paint number $\chi _{\text {DPP}}(G)$, the weak coloring number $\text {wcol}(G)$, the coloring number $\text {col}(G)$. Our result generalizes some corresponding known results.