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 $n\geqslant 10$ are characterized; as well 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. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.Comment: 25 pages, 5 figure

Five results on maximizing topological indices in graphs

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.Comment: 13 pages, 4 figure