  Graph operations or products, such as triangulation and Kronecker product
have been extensively applied to model complex networks with striking
properties observed in real-world complex systems. In this paper, we study
hitting times and resistance distances of $q$-triangulation graphs. For a
simple connected graph $G$, its $q$-triangulation graph $R_q(G)$ is obtained
from $G$ by performing the $q$-triangulation operation on $G$. That is, for
every edge $uv$ in $G$, we add $q$ disjoint paths of length $2$, each having
$u$ and $v$ as its ends. We first derive the eigenvalues and eigenvectors of
normalized adjacency matrix of $R_q(G)$, expressing them in terms of those
associated with $G$. Based on these results, we further obtain some interesting
quantities about random walks and resistance distances for $R_q(G)$, including
two-node hitting time, Kemeny's constant, two-node resistance distance,
Kirchhoff index, additive degree-Kirchhoff index, and multiplicative
degree-Kirchhoff index. Finally, we provide exact formulas for the
aforementioned quantities of iterated $q$-triangulation graphs, using which we
provide closed-form expressions for those quantities corresponding to a class
of scale-free small-world graphs, which has been applied to mimic complex
networks.
