The Harmonic index–radius conjecture

Let $G$ be a simple undirected connected graph with vertex set $V(G)=\{v_1,\dots,v_n\}$ and edge set $E(G)$. Let $d_i$ be the degree of $v_i$, define the Harmonic index by

$$H(G)=\sum_{v_i v_j\in E(G)}\frac{2}{d_i+d_j},$$

and let $r(G)$ be the graph radius.

Harmonic index–radius conjecture. For all connected graphs $G$ except even paths,

$$H(G)\geq r(G).$$

The conjecture is proposed as an analogue of the strengthened Randić index–radius conjecture. The paper proves related bounds in terms of the cyclomatic number, including the asserted inequality for graphs with cyclomatic number at most four, but leaves the general statement open.