Wiener index in graphs with given minimum degree and maximum degree

TL;DR

This paper improves bounds on the Wiener index for graphs with specified minimum and maximum degrees, providing sharper estimates especially for graphs with large maximum degree and specific forbidden subgraphs.

Contribution

It introduces a significantly improved asymptotic upper bound on the Wiener index considering both minimum and maximum degrees, including special cases for triangle-free and C4-free graphs.

Findings

New asymptotically sharp bound for Wiener index with large maximum degree

Bound for triangle-free graphs derived and validated

Optimal bound for C4-free graphs established

Abstract

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.