Spectra, Hitting Times and Resistance Distances of q- Subdivision Graphs. (9th December 2019) - Zeng, Yibo; Zhang, Zhongzhi
Abstract: Subdivision, triangulation, Kronecker product, corona product and many other 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.