The Relation Between the Harmonic Index and Some Coloring Parameters

Let H(G) be the harmonic index of a graph G, which is defined as: \(\begin{aligned} H(G) = \sum _{uv \in E(G)}\frac{2}{d_{G}(u) + d_{G}(v)}. \end{aligned}\) $$\begin{array}{c}\hfill H(G)=\sum _{uv\in E(G)}\frac{2}{d_G(u)+d_G(v)}.\end{array}$$ In this note, we define a new graph parameter \(\xi (G)\) $\xi (G)$ satisfying some properties and prove that \(\xi (G) \le 2H(G)\) $\xi (G)\le 2H(G)$ , with equality if and only if G is a non-trivial complete graph, possibly plus some additional isolated vertices. In particular, \(\xi (G)\) $\xi (G)$ can be the chromatic number \(\chi (G)\) $\chi (G)$ , the choice number \(\chi _{\ell }(G)\) ${\chi }_{\ell }(G)$ , the DP-chromatic number \(\chi _{\text {DP}}(G)\) ${\chi }_{\text{DP}}(G)$ , the DP-paint number \(\chi _{\text {DPP}}(G)\) ${\chi }_{\text{DPP}}(G)$ , the weak coloring number \(\text {wcol}(G)\) $\text{wcol}(G)$ , the coloring number \(\text {col}(G)\) $\text{col}(G)$ . Our result generalizes some corresponding known results.