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, ...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.展开更多
In this paper we give six explicit formulae to compute the Kirchhoff index,the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index of the k-cactus chain and the cactus graph which can be obta...In this paper we give six explicit formulae to compute the Kirchhoff index,the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index of the k-cactus chain and the cactus graph which can be obtained from a k-cactus chain by expanding each of the cut-vertices to a cut edge.展开更多
文摘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.
基金Supported by the National Natural Science Foundations of China(No.11401102)
文摘In this paper we give six explicit formulae to compute the Kirchhoff index,the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index of the k-cactus chain and the cactus graph which can be obtained from a k-cactus chain by expanding each of the cut-vertices to a cut edge.
