On Topological Indices and Their Reciprocals

If TI(G) = Sigma(gamma) F(gamma) is a topological index of the graph G, then RTI(G) = Sigma(gamma) 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 vPertices u and v, then the Wiener and Harary indices are W = Sigma(u,v) d(u, v) and H = Sigma(u,v) 1/d(u,v), respectively. In this paper the product W center dot H is studied. The minimum value of W center dot H is determined for general connected graphs and conjectured for trees. The maximum value is discussed, based on our computer-aided findings.