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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.12781 |
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| _version_ | 1866917987988013056 |
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| author | Li, Hao Chen, Xinyi Liu, Hao |
| author_facet | Li, Hao Chen, Xinyi Liu, Hao |
| contents | Substituting each edge of a simple connected graph $G$ by a path of length 1 and $k$ paths of length 5 generates the $k$-hexagonal graph $H^k(G)$. Iterative graph $H^k_n(G)$ is produced when the preceding constructions are repeated $n$ times. According to the graph structure, we obtain a set of linear equations, and derive the entirely normalized Laplacian spectrum of $H^k_n(G)$ when $k = 1$ and $k \geqslant 2$ respectively by analyzing the structure of the solutions of these linear equations. We find significant formulas to calculate the Kemeny's constant, multiplicative degree-Kirchhoff index and number of spanning trees of $H^k_n(G)$ as applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hexagonal and k-hexagonal graph's normalized Laplacian spectrum and applications Li, Hao Chen, Xinyi Liu, Hao Combinatorics Substituting each edge of a simple connected graph $G$ by a path of length 1 and $k$ paths of length 5 generates the $k$-hexagonal graph $H^k(G)$. Iterative graph $H^k_n(G)$ is produced when the preceding constructions are repeated $n$ times. According to the graph structure, we obtain a set of linear equations, and derive the entirely normalized Laplacian spectrum of $H^k_n(G)$ when $k = 1$ and $k \geqslant 2$ respectively by analyzing the structure of the solutions of these linear equations. We find significant formulas to calculate the Kemeny's constant, multiplicative degree-Kirchhoff index and number of spanning trees of $H^k_n(G)$ as applications. |
| title | Hexagonal and k-hexagonal graph's normalized Laplacian spectrum and applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.12781 |