Diameter of General Knödel Graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866917388011700224 |
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| author | Musawi, Seyed Reza Kiashi, Esameil Nazari |
| author_facet | Musawi, Seyed Reza Kiashi, Esameil Nazari |
| contents | The Knödel graph $W_{Δ,n}$ is a $Δ$-regular bipartition graph on $n\ge 2^Δ$ vertices and $n$ is an even integer. The vertices of $W_{Δ,n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le j\le n/2-1$. For every $j$, $0\le j\le n/2-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1) \mod (n/2))$, for $k=0,1,\cdots,Δ-1$. In this paper we obtain some formulas for evaluating the distance of vertices of the Knödel graph and by them, we provide the formula $diam(W_{Δ,n})=1+\lceil\frac{n-2}{2^Δ-2}\rceil$ for the diameter of $W_{Δ,n}$, where $n\ge (2Δ-5)(2^Δ-2)+4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_05435 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Diameter of General Knödel Graphs Musawi, Seyed Reza Kiashi, Esameil Nazari Combinatorics 05C12, 05C30, 05C38 The Knödel graph $W_{Δ,n}$ is a $Δ$-regular bipartition graph on $n\ge 2^Δ$ vertices and $n$ is an even integer. The vertices of $W_{Δ,n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le j\le n/2-1$. For every $j$, $0\le j\le n/2-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1) \mod (n/2))$, for $k=0,1,\cdots,Δ-1$. In this paper we obtain some formulas for evaluating the distance of vertices of the Knödel graph and by them, we provide the formula $diam(W_{Δ,n})=1+\lceil\frac{n-2}{2^Δ-2}\rceil$ for the diameter of $W_{Δ,n}$, where $n\ge (2Δ-5)(2^Δ-2)+4$. |
| title | Diameter of General Knödel Graphs |
| topic | Combinatorics 05C12, 05C30, 05C38 |
| url | https://arxiv.org/abs/2004.05435 |