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

TL;DR

This paper characterizes graphs with extreme Wiener indices among unicyclic and bicyclic graphs, and studies the ratios between Szeged indices and Wiener index, providing bounds and identifying specific graphs with extremal ratios.

Contribution

It determines extremal Wiener indices for certain classes of graphs and establishes bounds on the ratios of Szeged indices to Wiener index, including identifying graphs with extremal ratios.

Findings

Characterized graphs with the 4th to 7th largest Wiener indices among unicyclic graphs.

Identified graphs with the top four largest Wiener indices among bicyclic graphs.

Established sharp lower bounds on the ratio Sz(G)/W(G) for connected graphs with non-complete blocks.

Abstract

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.