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

Let H be a graph with vertex set V(H)=₁, v₂, ⋯, vₖ\. The generalized blow-up graph Hp₁,…,pₖq₁,…,qₖ is constructed by replacing each vertex vᵢ ∈ V(H) with the graph Gᵢ = pᵢKₜ ∪ qᵢK₁(i=1,2,⋯,k)$, then connecting all vertices between $G_i$ and $G_j$ whenever $v_iv_j ∈ E(H)$. In this paper, we enumerate the spanning trees in generalized blow-up graphs $Hp_1, p_2, ⋯, p_kq_1, q_2, ⋯, q_k$, 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 Hp₁, p₂, ⋯, pₖq₁, q₂, ⋯, qₖ, 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].