Abstract
If \( TI(G)=\sum\limits_\gamma F(\gamma) \) is a topological index of the graph \( G \),
then \( RTI(G)=\sum\limits_\gamma \frac{1}{F(\gamma)} \) is the respective reciprocal index.
In contemporary mathematical chemistry, a large number of pairs \( (TI,RTI) \) have been
separately introduced and studied, but their mutual relations eluded attention.
In this paper, we determine some basic relations between \( TI \) and \( RTI, \) and then
focus our attention to the pair Wiener index - Harary index.
If \( G \) is a connected graph and \( d(u,v) \) the distance between its vertices \( u \) and \( v \),
then the Wiener and Harary indices are \( W = \sum\limits_{u,v} d(u,v) \) and
\( H = \sum\limits_{u,v} \frac{1}{d(u,v)} \), respectively. In this paper the product \( W\!\cdot\!H \)
is studied. The minimum value of \( W\!\cdot\!H \) is determined for general connected
graphs and conjectured for trees. The maximum value is discussed, based on our
computer-aided findings.
