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.
Highlights
The Wiener index was originally introduced as a tool in chemistry, while the average distance is a useful tool for the analysis of networks in mathematics and computer science since it is an indicator for the expected distance between two randomly chosen points in a network
The graphs that demonstrate that 1 is sharp apart from an additive constant are either regular or close to regular, and this applies to the corresponding bounds for triangle-free graphs in Dankelmann and Entringer (2000). This suggests that stronger bounds hold for graphs of given minimum degree that contain a vertex whose degree is significantly larger, for example if the maximum degree is cn, where c ∈ (0, 1) and n is the order of the graph
The aim of this paper is to show that this is the case and to determine upper bounds on the Wiener index or average distance of graphs in terms of order, minimum degree and maximum degree that are sharp apart from an additive constant
Summary
The graphs that demonstrate that 1 is sharp apart from an additive constant are either regular or close to regular, and this applies to the corresponding bounds for triangle-free graphs in Dankelmann and Entringer (2000) This suggests that stronger bounds hold for graphs of given minimum degree that contain a vertex whose degree is significantly larger, for example if the maximum degree is cn, where c ∈ (0, 1) and n is the order of the graph. The aim of this paper is to show that this is the case and to determine upper bounds on the Wiener index or average distance of graphs in terms of order, minimum degree and maximum degree that are sharp apart from an additive constant.
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