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 andWiener 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$n\geqslant 10$ are characterized; as well the graphs with the first, second,third, and fourth largest Wiener indices among all bicyclic graphs areidentified. Based on these results, further relation on the quotients betweenthe (revised) Szeged index and the Wiener index are studied. Sharp lower boundon $Sz(G)/W(G)$ is determined for all connected graphs each of which containsat least one non-complete block. As well the connected graph with the secondsmallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least onecycle.Comment: 25 pages, 5 figure