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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2406.04060 |
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| _version_ | 1866916295182647296 |
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| author | Sun, Wensheng Yang, Yujun Chen, Wuxian Xu, Shou-Jun |
| author_facet | Sun, Wensheng Yang, Yujun Chen, Wuxian Xu, Shou-Jun |
| contents | Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. The resistance diameter of $G$, denoted by $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solution to a conjecture on resistance distances of block tower graphs Sun, Wensheng Yang, Yujun Chen, Wuxian Xu, Shou-Jun Combinatorics Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. The resistance diameter of $G$, denoted by $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture. |
| title | Solution to a conjecture on resistance distances of block tower graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.04060 |