Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.08841 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912640348979200 |
|---|---|
| author | Mallu, Sufiyan |
| author_facet | Mallu, Sufiyan |
| contents | Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( ρ(D) \) of \( D \) is the maximum of the average distances of the vertices in \( D \).
In this paper, we provide a sharp upper bound on the remoteness of a strong digraph with given order, size, and vertex-connectivity. We then characterise the extremal digraphs that maximise remoteness among all strong digraphs of order \(n\), size at least \(m\), and vertex-connectivity \(κ\). Finally, we demonstrate that the upper bounds on the remoteness of a graph given its order, size, and connectivity constraints (see \cite{DanMafMal2025}) can be extended to a larger class of digraphs containing all graphs, the Eulerian digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remoteness, order, size and connectivity constraints in digraphs Mallu, Sufiyan Combinatorics 05C12 Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( ρ(D) \) of \( D \) is the maximum of the average distances of the vertices in \( D \). In this paper, we provide a sharp upper bound on the remoteness of a strong digraph with given order, size, and vertex-connectivity. We then characterise the extremal digraphs that maximise remoteness among all strong digraphs of order \(n\), size at least \(m\), and vertex-connectivity \(κ\). Finally, we demonstrate that the upper bounds on the remoteness of a graph given its order, size, and connectivity constraints (see \cite{DanMafMal2025}) can be extended to a larger class of digraphs containing all graphs, the Eulerian digraphs. |
| title | Remoteness, order, size and connectivity constraints in digraphs |
| topic | Combinatorics 05C12 |
| url | https://arxiv.org/abs/2510.08841 |