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

TL;DR

This work studies generalized blow-up graphs $H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k}$, formed by replacing each base vertex with $G_i = p_iK_t \cup q_iK_1$ and connecting across edges, and seeks both spanning-tree counts and resistance-distance metrics. It derives a compact closed form for the number of spanning trees in terms of $n_i = t p_i + q_i$ and $n = \sum_i n_i$, generalizing previous results through Laplacian-eigenvalue techniques and network substitutions. For resistance distances and Kirchhoff indices, the paper provides explicit, case-based formulas via an equivalent weighted network $G^*$ built from $H\star(n_1,\ldots,n_k)^{\omega}$, introducing eight distance parameters $r_1,\ldots,r_8$ to capture intra- and inter-block relations, and yielding a comprehensive expression for $Kf(H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k})$. The results extend several strands of prior work to a broad generalized blow-up setting and include corollaries for unbalanced blow-ups and core-satellite configurations, with potential implications for network design and analysis in terms of spanning-tree enumeration and effective resistance.

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].

Figures (11)

• Figure 1: An example for the blow-up transformation.
• Figure 2: The Star-Triangle Transformation.
• Figure 3: The Mesh-star transformation.
• Figure 4: The $K_{m,n}$-double star transformation with edge weight.
• Figure 5: The $K_{m,n}$-double star transformation with vertex weight.

Theorems & Definitions (28)

• Proposition 2.1: Principle of Elimination, Klein kled2002
• Proposition 2.2: $S$-Equivalent Networks
• Proposition 2.3: Principle of Substitution, Gervacio gerv2016
• Remark 2.1
• Proposition 2.4: Star-Triangle Transformation, Kennelly kenn1899
• Lemma 2.1: Li and Tian liti2022
• Lemma 2.2: Chen and Yan cwya2021
• Lemma 2.3: Klein and Randić klei1993
• Lemma 3.1: Biggs bigg1974
• Theorem 3.1