On $\ell$-distance-balancedness of cubic Cayley graphs of dihedral groups
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912170055303168 |
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| author | Ma, Gang Wang, Jianfeng Li, Guang Klavžar, Sandi |
| author_facet | Ma, Gang Wang, Jianfeng Li, Guang Klavžar, Sandi |
| contents | A connected graph $Γ$ of diameter ${\rm diam}(Γ) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(Γ)|=|W_{yx}(Γ)|$ for every $x,y\in V(Γ)$ with $d_Γ(x,y)=\ell$, where $W_{xy}(Γ)$ is the set of vertices of $Γ$ that are closer to $x$ than to $y$. $Γ$ is said to be highly distance-balanced if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(Γ)]$. It is proved that every cubic Cayley graph whose generating set is one of $\{a,a^{n-1},ba^r\}$ and $\{a^k,a^{n-k},ba^t\}$ is highly distance-balanced. This partially solves a problem posed by Miklavič and Šparl. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $\ell$-distance-balancedness of cubic Cayley graphs of dihedral groups Ma, Gang Wang, Jianfeng Li, Guang Klavžar, Sandi Combinatorics A connected graph $Γ$ of diameter ${\rm diam}(Γ) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(Γ)|=|W_{yx}(Γ)|$ for every $x,y\in V(Γ)$ with $d_Γ(x,y)=\ell$, where $W_{xy}(Γ)$ is the set of vertices of $Γ$ that are closer to $x$ than to $y$. $Γ$ is said to be highly distance-balanced if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(Γ)]$. It is proved that every cubic Cayley graph whose generating set is one of $\{a,a^{n-1},ba^r\}$ and $\{a^k,a^{n-k},ba^t\}$ is highly distance-balanced. This partially solves a problem posed by Miklavič and Šparl. |
| title | On $\ell$-distance-balancedness of cubic Cayley graphs of dihedral groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.18893 |