maximizing the inverse degree in a graph

Question

The inverse degree in the graph $G$ is defined as \begin{align*} r(G) = \sum_{i=1}^N \frac{1}{d_i}, \end{align*} where $d_i$ is the degree of node (vertex) $i$. Is the connected graph with maximum $r(G)$ always of a diameter $1$ or $2$ for any fixed number of links (edges) $L \in [N-1, \binom{N}{2}]$?