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has been cited by the following article:

• TITLE: The Normalized Laplacians on Both Two Iterated Constructions Associated with Graph and Their Applications

  AUTHORS: Chang Liu, Yingui Pan, Jianping Li, Li Dai

  KEYWORDS: Normalized Laplacian, Multiplicative Degree-Kirchhoff Index, Kemeny’s Con-stant, Spanning Tree

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.8 No.5, May 15, 2020

  ABSTRACT: Given a simple connected graph G, we consider two iterated constructions associated with G: Fk (G) and Rk (G) . In this paper, we completely obtain the normalized Laplacian spectrum of Fk (G) and Rk (G) , with k ≥2, respectively. As applications, we derive the closed-formula of the multiplicative degree-Kirchhoff index, the Kemeny’s constant, and the number of spanning trees of Fk(G) , Rk(G) , r-iterative graph ,Frk(G) and r-iterative graph , where k≥2 and r≥1 . Our results extend those main results proposed by Pan et al. (2018), and we provide a method to characterize the normalized Laplacian spectrum of iteratively constructed complex graphs.