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.