Enumeration of spanning trees and resistance distances of generalized blow-up graphs

Abstract

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].