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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.15314 |
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| _version_ | 1866916701457612800 |
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| author | Liu, Hechao Li, Lu You, Lihua Hua, Hongbo Chen, Liang |
| author_facet | Liu, Hechao Li, Lu You, Lihua Hua, Hongbo Chen, Liang |
| contents | Let $H$ be a graph with vertex set $V(H)=\{v_1, v_2, \cdots, v_k\}$. The generalized blow-up graph $H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k}$ is constructed by replacing each vertex $v_i \in V(H)$ with the graph $G_i = p_iK_t \cup q_iK_1$$(i=1,2,\cdots,k)$, then connecting all vertices between $G_i$ and $G_j$ whenever $v_iv_j \in E(H)$.
In this paper, we enumerate the spanning trees in generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Ge [Discrete Appl. Math. 305 (2021) 145-153], Cheng, Chen and Yan [Discrete Appl. Math. 320 (2022) 259-269]. Furthermore, we determine the resistance distances and Kirchhoff indices of generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Sun, Yang and Xu [Discrete Math. 348 (2025) 114327], Xu and Xu [Discrete Appl. Math. 362 (2025) 18-33], Ni, Pan and Zhou [Discrete Appl. Math. 362 (2025) 100-108]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration of spanning trees and resistance distances of generalized blow-up graphs Liu, Hechao Li, Lu You, Lihua Hua, Hongbo Chen, Liang Combinatorics Let $H$ be a graph with vertex set $V(H)=\{v_1, v_2, \cdots, v_k\}$. The generalized blow-up graph $H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k}$ is constructed by replacing each vertex $v_i \in V(H)$ with the graph $G_i = p_iK_t \cup q_iK_1$$(i=1,2,\cdots,k)$, then connecting all vertices between $G_i$ and $G_j$ whenever $v_iv_j \in E(H)$. In this paper, we enumerate the spanning trees in generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Ge [Discrete Appl. Math. 305 (2021) 145-153], Cheng, Chen and Yan [Discrete Appl. Math. 320 (2022) 259-269]. Furthermore, we determine the resistance distances and Kirchhoff indices of generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Sun, Yang and Xu [Discrete Math. 348 (2025) 114327], Xu and Xu [Discrete Appl. Math. 362 (2025) 18-33], Ni, Pan and Zhou [Discrete Appl. Math. 362 (2025) 100-108]. |
| title | Enumeration of spanning trees and resistance distances of generalized blow-up graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.15314 |