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

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


Introduction
Graph matrices, such as adjacency, incident, Laplacian and normalized Laplacian matrices, can well describe the structure and complex dynamic informations of complex networks. The eigenvalues and eigenvectors of these matrices represent some significant physical or chemical properties of networks. Recently, the normalized Laplacian has been a research hotpot, due to the consistency of eigenvalues in spectral geometry and random processes. Moreover, iteratively constructed graphs are very common in complex networks. Therefore, how to characterize the normalized Laplacian spectrum of such graphs is still a question worthy of study. Laplacian matrix of G. Recent research on Laplace spectrum one can refer to [1] [2].
In 2007, Chen and Zhang [3] introduced a new resistance distance-based parameter, named the multiplicative degree-Kirchhoff index, defined as The multiplicative degree-Kirchhoff index is closely related to the normalized Laplacian matrix ( ) G  , which is defined as Hunter (2014) [5] studied the Kemeny's constant of G, which is denoted by ( ) Ke G . The Kemmeny's constant provides an interesting quantity for finite ergodic Markov chains and can be given by In recent years, more and more researchers have been devoted to studying the normalized Laplacian spectra. Relevant research results on normalized Laplacian and multiplicative degree-Kirchhoff index one may refer to [6]- [11]. Now we consider two iterated constructions associated with G: Motivated by [7] [10], in this paper, we completely obtain the normalized Laplacian spectrum of ( )

Main Results
In this section, we will give the main conclusions of this paper. Theorem 2.1. Let G be a simple connected graph with n vertices and m edges.
Then the normalized Laplacian eigenvalues of ( ) k F G ( 2 k ≥ ) can be obtained as following: (i) If λ is an eigenvalue of ( ) σ σ σ and 4 σ are the eigenvalues of ( ( )) Then the normalized Laplacian eigenvalues of ( ) k R G ( 2 k ≥ ) can be obtained as following: µ are the eigenvalues of ( ( )) (ii) 0,

Preliminaries
Before you begin to format your paper, first write and save the content as a separate text file. Let G be a connected graph with n vertices and m edges, the order . Similarly, the order of ( ) k R G is 4 n km + and its size is (6 1) k m + . Let , is an n m × matrix. The rank of the directed incident matrix ( ) I G  satisfies the following lemma.
is an eigenvalue of ( ) G  with the same multiplicity as that of σ .

Proof. Let
, k 3-cycles are added between the vertices u, v. Denote these 3-cycles by be an eigenvector with respect to the eigenvalue we can obtain that for any vertex For any vertex u V ∈ , let ( ) G N u be the set of the neighbour of vertex u inherited from G. According to the construction of ( ) k F G from G and (3.2) and (3.3), we can get Similarly, for any Analogously, for the vertex Combining (3.5)-(3.7), we can have

8) Journal of Applied Mathematics and Physics
Similarly, According to (3.8), (3.9), we can get that The eigenvector x can be completely decided by 0 ( ) t Then we have 3 2 (18 17 92 37) 0 combining with (3.11), it is easy to conclude that the above equation does not hold. Thus, This completes the proofs.
Lemma 3.7. Let µ be an eigenvalue of ( ( )) is an eigenvalue of ( ) G  with the same multiplicity as that of µ . Then we let 2,

Proof. Let
(3.14) Suppose 0 is an eigenvector with respect to the eigen- For any vertex u O ∈ , let ( ) G N u be the set of the neighbour of vertex u inherited from G. According to the structural of ( ) k R G from G, (3.14) and (3.15), we can get Similarly, for any Analogously, for the vertex For the vertices In view of the formulas (3.18)-(3.19), one can see that According to (3.21) and (3.22), we have Substituting (3.21) into (3.16), then we have is one of the corresponding eigenvectors. Hence, Since the connected simple graph G is non-bipartite, G contains an odd cycle, written by 0 C . By   This completes the proofs of (iv).
Then we have
(iv) Substituting Since the connected simple graph G is non-bipartite, G contains an odd cycle, written as 1 C . In view of (4.15), we have . It is easy to conclude that 0 . Combining with (3.16) to (3.20), we have  This completes the proofs of (iv).
Let Then the following equations system can be obtained.       By Vieta theorem, we obtain the following equation: