The quotients between the (revised) Szeged index and Wiener index of graphs

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on Sz(G)/W (G) is determined for all connected graphs each of which contains at least one non-complete block. In addition the connected graph with the second smallest value on Sz(G)/W (G) is identified for G containing at least one cycle.