Hitting times and resistance distances of q-triangulation graphs: Accurate results and applications
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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.
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