Spectra, hitting times, and resistance distances of $q$-subdivision graphs

TL;DR

This paper derives explicit formulas for spectral, resistance, and hitting time properties of $q$-subdivision graphs, revealing their structural characteristics and applications in modeling complex, scale-free fractal networks.

Contribution

It provides new explicit formulas for key graph invariants of $q$-subdivision graphs in terms of the original graph, including eigenvalues, hitting times, and resistance distances.

Findings

Explicit formulas for eigenvalues and eigenvectors of $S_q(G)$

Closed-form expressions for resistance distances and Kirchhoff indices

Analysis of iterated $q$-subdivision graphs for hierarchical lattices

Abstract

Graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its $q$-subdivision graph $S_q(G)$ is obtained from $G$ through replacing every edge $uv$ in $G$ by $q$ disjoint paths of length 2, with each path having $u$ and $v$ as its ends. We derive explicit formulas for many quantities of $S_q(G)$ in terms of those corresponding to $G$, including the eigenvalues and eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny constant, two-node resistance distance, Kirchhoff index, additive degree-Kirchhoff index, and multiplicative degree-Kirchhoff index. We also study the properties of the iterated $q$-subdivision graphs, based on which we obtain the closed-form expressions for a family of hierarchical lattices, which has been used to describe scale-free fractal networks.