Properties and Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

We study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and show some applications. We obtain some results on the Laplacian spectral radius of a graph by grafting and adding edges. We also determine the structure of the maximal Laplacian spectrum tree among trees with n vertices and k pendant vertices (n, k fixed), and the upper bound of the Laplacian spectral radius of some trees.


Introduction
The theory of graph spectra has been established in the 1950s and 1960s.Chung has taken the investigation of the theory of graph spectra to a new level by a 45-minute report [1] presented at the World Congress of Mathematicians in 1994, and in monographs [2].Applications of the theory of graph spectra have also been found in the fields of electrical networks and vibration theory [3,4].The wide range of the applications of the theory of graph spectra has led it to become a very active field of research of graph theory over the last thirty to forty years, and large numbers of results are continuously emerging.
There are many results on the (adjacency) spectral radius for different classes of graphs.Guo et al. [5] have studied the largest and the second largest spectral radius of trees with  vertices and diameter .Guo and Shao [6] have studied the first ⌊/2⌋ + 1 (where ⌊⌋ represents the maximal integer not more than ) spectral radii of graphs with  vertices and diameter .Wu et al. [7] have studied the spectral radius of trees on  pendant vertices.
There are also many results on the Laplacian spectral radius for different classes of graphs.Hong and Zhang [8] have studied the upper and lower bounds for the Laplacian spectral radius of trees.Guo [9] has studied the second largest Laplacian eigenvalues of trees.In the paper, we further study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and obtain some applications on the Laplacian spectral radius of trees with  pendant vertices.
In Section 2, we describe some properties of the eigenvector corresponding to the Laplacian spectral radius of a graph.Section 3 presents some applications of the eigenvector corresponding to the Laplacian spectral radius of a graph, including some results on the Laplacian spectral radius of a graph by grafting and adding edges.Then we obtain the structure of the maximal Laplacian spectrum tree among trees with  vertices and  pendant vertices (,  fixed), and the upper bound of the Laplacian spectral radius for some trees.

Properties of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph
For convenience, we denote () by  sometimes.Since the Laplacian matrix () is a real symmetric matrix, we have the following theorem.
Proof.(1) It is proved by the definition.
Proof.(1) Since  is a real symmetric matrix, we have () ∈ .Thus,  is a real vector.

Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius
In this section, we present some applications of the eigenvector of the Laplacian spectral radius of a graph.Theorem 6.Let  and V be the vertices of tree T. Denote the distance of vertices  and V by (, V) and (, V) = , where  is an odd number and  ≥ 3. Let   be the graph obtained from  by adding edge (, V).Then (  ) > ().
According to the well-known Courant-Weyl inequalities, we have the following lemma.
Lemma 7 (see [15]).Let  be a graph with  vertices and let   =  +  be the graph obtained by inserting a new edge  into .Let   () and   (  ) be the eigenvalues of the Laplacian matrix of  and   ( = 1, 2, . . ., ), respectively.
Lemma 8 (see [16]).Let  be a connected graph for  vertices with at least one edge.Then () ≥ Δ() + 1, where Δ() is the largest degree of the graph , and equality holds if and only if Δ() =  − 1.
Proof.We have to prove that if  ∈ T , , then () ≤ ( , ) with equality only when  =  , .Let  be the cardinality of the vertices of degree 3 or greater.
Proof.The proof is obvious from Lemma 13.
For the sake of clarity, we identify a graph by its characteristic polynomial.Let Φ(()) be the characteristic polynomial of graph .For convenience, we denote Φ(()) by Φ() sometimes.First, we require some lemmas.
Lemma 16 (see [9]).Let  = V be a cut edge of the simple connected graph . 1 and  2 are two connected branches of  − V, where  ∈ ( 1 ), Here, Φ( V ( 2 )) denotes the determinant obtained by deleting the row and column for vertex V of  2 from the determinant for ( 2 ), and Φ(  ( 1 )) denotes the determinant obtained by deleting the row and column for vertex  of  1 from the determinant for ( 1 ).
Lemma 17 (see [18]).Let   be a path with  vertices, let  0  be the path obtained by adding a loop on a pendant vertex in   , and let  00  be the path obtained by adding loops on the two pendant vertices in   .Suppose that the degree of contribution corresponding to each loop is 1 and Φ( 0 ) = 0, Φ( 0 0 ) = 1, Φ( 00 0 ) = 1.Then (20) In general, the Laplacian spectral radius is difficult to calculate although the characteristic polynomial can be identified by Chebvshev polynomials.Thus, we only give some results in some cases.