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

摘要

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.