Abstract
We mainly study the properties of the 4-double-star-like tree, which is the generalization of star-like trees. Firstly we use graft transformation method to obtain the maximal and minimum 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.
1. 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.
2. 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 denote the eigenvalues with . The sequence of eigenvalues of is named the spectrum of , and the spectral radius of often calls the largest eigenvalue and is denoted by . The characteristic polynomial of is called the characteristic polynomial of the graph and is denoted by . When is connected, is an irreducible matrix. And by Perron-Frobenius theorem [12], is simple and the correspondent eigenvector can be positive. We refer to such eigenvector as Perron vector of .
An internal path in a graph, denoted by , is a path joining vertices and 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 . The degree sum of vertices adjacent to is called the second degree of , remarked as [4].
In this paper, we denote by and the cycle with order and the third small spectral radius of tree. denotes the family of the 4-double-star-like tree. Let denote the degree of . 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 thenwith 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.
3. 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 be an edge of a connected graph with order , and denote the graph obtained from by subdividing the edge by a new vertex . Then the following two properties hold:(1)If is not on any internal path of , then .(2)If is on an internal path of , then .
Lemma 2 (see [15]). Let be a vertex of a connected graph with and, for positive integers and , let denote the graph got from by adding two pendent paths of length and at . If , then .
Lemma 3 (see [15]). Let be a connected graph with and and , and let be the graph obtained from by adding two pendent paths of and at . If and , then and .
From the above lemma we immediately get the following.
Corollary 4. Let be a 4-double-star-like tree. As shown in Figure 1, when two centers are adjacent, is denoted by with and , where denote the length of the 6 branches of the 4-double-star-like tree in clockwise direction, respectively. Then, by the graft transformation, we get a graph, as shown in Figure 2 denoted by . Then
Lemma 5 (see [1]). Let be a connected graph with order and spectral radius and denote its adjacency matrix. Let be any polynomial. ThenFurthermore, if the sum of all entries in each row of are not all the same, then the inequalities are strict.
The following result provides a very effective method which can affect the value of spectral radius.
Lemma 6 (see [2]). Suppose are two vertices of a graph , , and is the Perron vector of , where corresponds to the vertex . Let be the graph obtained from by deleting the edges and adding the edges . If , then .
The processes depicted in Lemma 6 are called graft transformation.
4. Main Results
In this section, we consider 4-double-star-like tree with vertices. The main result of the paper is the following theorem.
Theorem 7. The minimal extremal graph of 4-double-star-like tree is (as shown in Figure 3); namely,
Proof. Let , as shown in Figure 4. By Lemma 2, we are using the graft transformation to obtain the graph, as shown in Figure 5, denoted by , and, then, Suppose , , and and , , and are positive integers. For the convenience of proof, we use and to denote and , respectively. , , and denote the number of the vertices of three parts as shown in Figure 4. Similarly, , , and play the same role in Figure 5.
Obviously, is a subdivision of external of , and then, by using Lemma 1, we haveAs shown in Figure 6, is a subdivision of internal , and, by Lemma 1, we haveBy (7) and (8),Proceeding in this way, we haveThusAs shown in Figure 7, is the minimum extremal graphs of 4-double-star-like tree, and , shortened to .
Theorem 8. The maximum extremal graph of 4-double-star-like tree is (as shown in Figure 8), where are almost equal in length; namely, the length of each branch differs at most 1:
Proof. Let , as shown in Figure 9.
Obviously, is a subdivision of internal , and, by Lemma 1, we can getThen, as shown in Figure 10, is a subdivision of external , and, by Lemma 1, we can get By (13) and (14), we can getProceeding in this way until , we get , which can be denoted in the form of as in Corollary 4. The value of spectral radius is largest when equal and are almost equal in length, and then we get , as shown in Figure 11.
When , there are 6 different 4-double-star-like trees (as shown in Figure 12). By calculating the spectral radius of the 6 graphs, we can see that Theorem 8 holds for the 6 graphs:and then
Theorem 9. The upper bound of spectral radius of the 4-double-star-like trees is ; namely,
Proof. By the definition of 4-double-star-like tree we can getLet be the adjacent matrix of . Since the sum of entries in the th row of is , . Since the sum of entries in the th row of is , .
By (19) we obtainSuppose . By Lemma 5, we haveThusNow, we complete the proof of Theorem 9.
5. Conclusion
In this paper we have shown some results of the 4-double-star-like trees. We obtain the minimal and the maximum extremal graphs of 4-double-star-like tree being and , and then we get the upper bound of spectral radius of the 4-double-star-like trees, which is .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (11426125 and 61473139), Program for Liaoning Excellent Talents in University LR2014016, and the Educational Commission of Liaoning Province (L2014239).