The Spectral Radius for a Class of Double-Star-Like Tree Systems with Maximal Degree 4

We mainly study the properties of the 4-double-star-like tree, which is the generalization of star-like trees. Firstly we use graft transformationmethod to obtain themaximal andminimum extremal graphs of 4-double-star-like trees. Secondly, by the relations between the degree and second degree of vertices in maximal extremal graphs of 4-double-star-like trees we get the upper bounds of spectral radius of 4-double-star-like trees.


Introduction
In recent years, there are a lot of the results of the upper and lower bounds on the spectral radius of graphs.For example, Ellingham and Zha [1] proposed upper bound on the spectral radius of graphs embeddable on a given compact surface; Wang et al. [2] proved that  trees have the second and the third largest spectral radius; Gong [3] discussed some sharp lower bounds for spectral radius of connected graphs and gave a new lower bound for spectral radius; Yuan and Shu [4] gave upper bound for the spectral radius of Halin graphs; Zhou and Xu [5] showed the upper bounds for the Laplacian spectral radius of a graph by using the Cauchy-Schwarz inequality; Song and Wang [6,7] got the upper bound of the Laplacian spectral radius of some trees by grafting transformation.
Star-like tree and its generalization -star-like tree are special trees and widely used in computer science.There are a lot of results on the bounds of spectral radius of star-like trees and -star-like trees.For example, Patuzzi et al. [8] gave upper bounds of the spectral radius in star-like trees and double brooms; Aalipour et al. [9] obtained upper bound of double-star-like trees by Laplacian spectral; Wu and Hu [10] gave new upper bound of the spectral radii of -star-like trees.
Though many researchers obtained the bounds of spectral radius of star-like trees and -star-like trees, few used the extremal graphs of the two kinds of trees to get the bounds.In this paper, we mainly use graft transformation method to obtain the maximal and minimum extremal graphs of 4-double-star-like trees.Then by the relations between the degree and second degree of vertices in maximal extremal graphs of 4-double-star-like trees we get the upper bounds of spectral radius of 4-double-star-like trees.Ours result provide a new method to research the bound of spectral radius of star-like trees.By this new method, we may get the second largest index, the third largest index, or the th largest index of -star-like trees, which provide a valuable tool for further research on -star-like trees.

Preliminaries
In this paper, all graphs considered here are simple and undirected.We refer to [11] for notation not defined here.Let () be the adjacent matrix of graph .Since () is a real symmetric matrix, its eigenvalues must be real, and let  1 ,  2 , . . .,   denote the eigenvalues with  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   .The sequence of  eigenvalues of () is named the spectrum of , and the spectral radius of  often calls the largest eigenvalue  1 () and is denoted by ().The characteristic polynomial of () is called the characteristic polynomial of the graph  and is denoted by Φ().When  is connected, () is an irreducible matrix.And by Perron-Frobenius theorem [12], () is simple and the correspondent 2 Mathematical Problems in Engineering eigenvector can be positive.We refer to such eigenvector as Perron vector of .
An internal path in a graph, denoted by V 0 , V 1 , . . ., V  ( ≥ 1), is a path joining vertices V 0 and V  which are both of degree greater than 3, while all other vertices are of degree equal to 2.
If graph  with  vertices has exactly two vertices of degree greater than two, then  is double-star-like [13].A double-star-like tree graph of order  that has exactly two vertices with maximum degree 4 is called 4-double-star-like tree, and it is denoted by  2 4 ().The degree sum of vertices adjacent to V is called the second degree of V, remarked as (V  ) [4].
In this paper, we denote by   and  3 () the cycle with order  and the third small spectral radius of tree.Γ 2 4 () denotes the family of the 4-double-star-like tree.Let (V) denote the degree of V.The sum of entries in the th row of  is called   ().
Zhou and Xu [5] proved the following.
Let  = (, ) be a simple graph with  vertices and  edges, and then with equality if and only if  is a regular bipartite graph.Wu and Hu [10] gave a new upper bound of the spectral radii of -star-like trees.They show the following result: In this paper, we consider the extremal graphs and the upper bounds of the spectral radius of 4-double-star-like trees.The rest of the paper is organized as follows.Section 3 contains some useful lemmas which play a fundamental role.Section 4 contains three theorems on the 4-double-star-like trees.

Some Useful Lemmas
In this section, we use the following lemmas about the connected graph to prove our main results.
Lemma 1 (see [14]).Let V be an edge of a connected graph  with order , and  ,V denote the graph obtained from  by subdividing the edge V by a new vertex .Then the following two properties hold: (1) If V is not on any internal path of  ≇   , then ( ,V ) > ().
Lemma 2 (see [15]).Let V be a vertex of a connected graph  with (V) > 0 and, for positive integers  and , let  , denote the graph got from  by adding two pendent paths of length  and  at V. If  ≥  ≥ 1, then ( , ) > ( +1,−1 ).
From the above lemma we immediately get the following.
Corollary 4. Let  2 4 () be a 4-double-star-like tree.As shown in Figure 1, when two centers are adjacent, Lemma 5 (see [1]).Let  be a connected graph with order  and spectral radius  and  denote its adjacency matrix.Let  be any polynomial.Then r  q  p  Furthermore, if the sum of all entries in each row of () are not all the same, then the inequalities are strict.
The processes depicted in Lemma 6 are called graft transformation.