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Graph operations or products play an important role in complex networks. Inthis paper, we study the properties of $q$-subdivision graphs, which have beenapplied to model complex networks. For a simple connected graph $G$, its$q$-subdivision graph $S_q(G)$ is obtained from $G$ through replacing everyedge $uv$ 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 andeigenvectors of normalized adjacency matrix, two-node hitting time, Kemenyconstant, two-node resistance distance, Kirchhoff index, additivedegree-Kirchhoff index, and multiplicative degree-Kirchhoff index. We alsostudy the properties of the iterated $q$-subdivision graphs, based on which weobtain the closed-form expressions for a family of hierarchical lattices, whichhas been used to describe scale-free fractal networks.

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