On the number of spanning trees of bicirculant graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917213977444352 |
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| author | Yang, Jing Xian, Fangming |
| author_facet | Yang, Jing Xian, Fangming |
| contents | A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let
$Γ=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula
for the number of spanning trees of bicirculant graph $Γ$, investigate some arithmetic properties of the number of spanning trees of $Γ$, and find its asymptotic behaviour as $n$ tends infinity. In addition,
we show that $F(x)=\sum_{n=1}^{\infty}τ(Γ)x^n$ is a rational function with integer coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12899 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the number of spanning trees of bicirculant graphs Yang, Jing Xian, Fangming Combinatorics Spectral Theory 05C30 G.2.2 A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let $Γ=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of spanning trees of bicirculant graph $Γ$, investigate some arithmetic properties of the number of spanning trees of $Γ$, and find its asymptotic behaviour as $n$ tends infinity. In addition, we show that $F(x)=\sum_{n=1}^{\infty}τ(Γ)x^n$ is a rational function with integer coefficients. |
| title | On the number of spanning trees of bicirculant graphs |
| topic | Combinatorics Spectral Theory 05C30 G.2.2 |
| url | https://arxiv.org/abs/2601.12899 |