  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.
