Wiener index in graphs given girth, minimum, and maximum degrees

Authors

Fadekemi J. Osaye, Alabama State University, USAFollow
Liliek Susilowati, Airlangga University, Surabaya, IndonesiaFollow
Alex S. Alochukwu, Albany State University, USAFollow
Cadavious Jones, Rust College, USAFollow

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 $G$. 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 $\delta$ by Kouider and Winkler \cite{Kouider} was improved significantly by Alochukwu and Dankelmann \cite{Alex} for graphs containing a vertex of large degree $\Delta$ to $W(G) \leq {n-\Delta+\delta \choose 2} \big( \frac{n+2\Delta}{\delta+1}+4 \big)$. In this paper, we give upper bounds on the Wiener index of $G$ in terms of order $n$ and girth $g$, where $n$ is a function of both the minimum degree $\delta$ and maximum degree $\Delta$. Our result provides a generalisation for these previous bounds for any graph of girth $g$. In addition, we construct graphs to show that, if for given $g$, there exists a Moore graph of minimum degree $\delta$, maximum degree $\Delta$ and girth $g$, then the bounds are asymptotically sharp.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Osaye, Fadekemi J.; Susilowati, Liliek; Alochukwu, Alex S.; and Jones, Cadavious (2023) "Wiener index in graphs given girth, minimum, and maximum degrees," Theory and Applications of Graphs: Vol. 10: Iss. 2, Article 3.
DOI: 10.20429/tag.2023.10203
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol10/iss2/3