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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.19326 |
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| _version_ | 1866929562171998208 |
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| author | Mollard, Michel |
| author_facet | Mollard, Michel |
| contents | The Fibonacci-run graphs $\mathcal{R}_n$ are a family of an induced subgraph of hypercubes introduced by Eğecioğlu and Iršič in 2021.
A cyclic version of $\mathcal{R}_n$, the Lucas-run graph $\mathcal{R}_n^l$, was also recently proposed (Jianxin Wei, 2024). We prove that the generating function previously given for the polynomial $D_{\mathcal{R}_n}(x,q)$ which counts the number of hypercubes at a given distance in $\mathcal{R}_n$ was erroneous and determine its correct expression. We also consider Lucas-run graphs and prove the conjecture proposed by Jianxin Wei establishing the link between cube polynomials of $\mathcal{R}_n^l$ and $\mathcal{R}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distance cube polynomials of Fibonacci and Lucas-run graphs Mollard, Michel Combinatorics The Fibonacci-run graphs $\mathcal{R}_n$ are a family of an induced subgraph of hypercubes introduced by Eğecioğlu and Iršič in 2021. A cyclic version of $\mathcal{R}_n$, the Lucas-run graph $\mathcal{R}_n^l$, was also recently proposed (Jianxin Wei, 2024). We prove that the generating function previously given for the polynomial $D_{\mathcal{R}_n}(x,q)$ which counts the number of hypercubes at a given distance in $\mathcal{R}_n$ was erroneous and determine its correct expression. We also consider Lucas-run graphs and prove the conjecture proposed by Jianxin Wei establishing the link between cube polynomials of $\mathcal{R}_n^l$ and $\mathcal{R}_n$. |
| title | Distance cube polynomials of Fibonacci and Lucas-run graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.19326 |