Number of edges with shortest cycle k in a Kautz graph
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909898405576704 |
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| author | Faber, Vance |
| author_facet | Faber, Vance |
| contents | For the Kautz digraph $K(d,D)$, let $ρ_k(d,D)$ be the number of oriented edges whose shortest directed cycle has length $k+1$, and define $Δ_k(d,D) = ρ_k(d,D) - ρ_k(d,D-1)$. We give an exact, finite-dimensional matrix product that computes $Δ_k(d,D)$ directly, without first computing $ρ$. In particular,
$Δ_k(d,D)=0$ for $k < D/2+2$.
and $Δ_k(d,D)$ is positive for every larger $k$ up to $D-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Number of edges with shortest cycle k in a Kautz graph Faber, Vance Combinatorics 05C20, 05C38 For the Kautz digraph $K(d,D)$, let $ρ_k(d,D)$ be the number of oriented edges whose shortest directed cycle has length $k+1$, and define $Δ_k(d,D) = ρ_k(d,D) - ρ_k(d,D-1)$. We give an exact, finite-dimensional matrix product that computes $Δ_k(d,D)$ directly, without first computing $ρ$. In particular, $Δ_k(d,D)=0$ for $k < D/2+2$. and $Δ_k(d,D)$ is positive for every larger $k$ up to $D-1$. |
| title | Number of edges with shortest cycle k in a Kautz graph |
| topic | Combinatorics 05C20, 05C38 |
| url | https://arxiv.org/abs/2511.08385 |