Abstract

Let be an integer, let denote the Brualdi-Li matrix of order , and let denote the Laplacian matrices of Brualdi-Li tournament digraphs. We obtain the eigenvalues and eigenvectors of .

1. Introduction

The Laplacian spectral theory is currently not only a hot research direction of spectral graph theory but also one of the research directions of combined matrix theory. The Laplacian spectrum of a graph is of importance in graph theory, matrix theory, and the definite solution of partial differential equations. It also has applications in quantum chemistry, biology, and complex network. Therefore, it has important theoretical and practical values to study the Laplacian eigenvalues and eigenvectors of graphs. The Laplacian spectrum of graph has attracted the attention of researchers; see [15] and so on. We follow [1, 6] for terminology and notations.

Let be a digraph of order with vertex set and arc set , where . The adjacency matrix of is the matrix of order , where if there is an arc from to and otherwise. The digraph is called the associated digraph of matrix . Let , the diagonal matrix with vertex outdegrees of . is called the Laplacian matrix of the digraph . The characteristic polynomial of the adjacency matrix , that is, , is called the characteristic polynomial of the digraph . The equation has complex roots and these roots are called the eigenvalues of . Suppose the distinct eigenvalues of are denoted by with corresponding algebraic multiplicities , where is a nonnegative integer . is called the Laplacian spectrum of digraph . The Laplacian spectral radius of is the largest modulus of an eigenvalue of , denoted by . The symbol will denote the complex field. Let be an eigenvalue of matrix . There is vector satisfying , and then is called the eigenvectors of matrix corresponding to .

A tournament matrix of order is a matrix satisfying the equation , where is the all ones matrix, is the identity matrix, and is the transpose of . Let where is strictly upper triangular tournament matrix (all of whose entries above the main diagonal are equal to one); that is, is the tournament matrix of order .

The matrix has been dubbed by the Brualdi-Li matrix. The associated digraph of matrix is called the Brualdi-Li tournament digraph. In 1983 Brualdi and Li conjectured that the maximal spectral radius for tournaments of order is attained by the Brualdi-Li matrix [7]. This conjecture has been confirmed in [8]. The properties of the Brualdi-Li matrix have been investigated in [914].

In this paper, we obtain the spectrum and eigenvectors of the Laplacian matrices of the Brualdi-Li tournament digraphs.

Theorem 1. Let be an integer, and let be the Laplacian spectrum of the Brualdi-Li tournament digraph. Then where is the floor of number , , and is the conjugate complex number of , .

Theorem 2. Let be an integer, and let be the eigenvector of corresponding to , where , ; then(1)if , then is the eigenvector of corresponding to , is the eigenvector of corresponding to , is the eigenvector of corresponding to ,where is an arbitrary constant, , (2)if , then

Corollary 3. Let be an integer, and let be the Laplacian spectral radius of the Brualdi-Li tournament digraph. Then

Proof. By Theorem 1,

Corollary 4. Let be an integer, then(1)if is odd, then is not diagonalizable,(2)if is even or , then is diagonalizable.

2. Some Lemmas

Fundamental Theorem of Algebra. Every nonzero, single-variable, degree polynomial with complex coefficients has, counted with multiplicity, exactly roots.

Complex Conjugate Root Theorem. If is a polynomial in one variable with real coefficients and is a root of with and real numbers, then its complex conjugate is also a root of , where .

The symbol denotes the Laplacian matrix of the Brualdi-Li tournament digraph. Clearly, .

Lemma 5. Let be an integer, and , where is real variable. Then

Proof. Consider

Lemma 6. Let be an integer, , let be an arbitrary eigenvalue of , and let be the eigenvector of corresponding to , where , . Let , , and , where is real variable. Then where , .

Proof. If is an eigenvalue, with eigenvector , of , then expands to Therefore, We have According to Lemma 5, we have Notice that this equation holds for too. Consider By Cramer’s rule, We are done.

Lemma 7. Under the assumptions and in the notation of Lemma 6,

Proof. In Lemma 6 , by setting , we have Since , it follows that As we all know that the eigenvalues of a real skew-symmetric matrix are all pure imaginary or nonzero, hence , where . Since hence . Notice that and if , then and . By Lemma 6, for arbitrary real variable . It is not possible. Hence . It is easy to see that and .

Lemma 8 (see [15]). If is a square matrix, then,(1)for every eigenvalue of , the geometric multiplicity is less than or equal to the algebraic multiplicity,(2) is diagonalizable if and only if the geometric multiplicity of every eigenvalue is equal to the algebraic multiplicity.

3. Proof of Theorem 1

Let be an arbitrary eigenvalue of , and let be the eigenvector of corresponding to , where , .

For , then , . Obviously, , are eigenvalues of and . Theorem 1 holds.

For , using the previous assumptions and notation, obviously, , by definition, is an eigenvalue of , .

Note that , and by definition, is an eigenvalue of . By simple calculation, the rank of matrix is equal to . By Lemma 8    .

For , by Lemma 7, we set ; hence . In Lemma 6 , by setting , we have

Hence Denoting , , and , we have

It must be that

Therefore That is, .

Since , , , and , then Denote , . As , we have ; that is, , where .

Therefore, where , .

Note that if is odd, by , then ; hence . It is an eigenvalue of .

Furthermore, . Note that , .

To sum up, by fundamental theorem of algebra and complex conjugate root theorem, for , we obtain the following conclusions.If is odd, then and .If is even, then and .Consider  . We complete the proof of Theorem 1.

4. Proofs of Theorem 2 and Corollary 4

It is easy to see that the distinct eigenvalues of are , with corresponding algebraic multiplicities , .

For , and, obviously, is the eigenvector of corresponding to .

For ,

By simple calculation, is the eigenvector of corresponding to .

For , let be the eigenvector of corresponding to , and then expands to Equation (30) is equivalent to the following equation:

By simple calculation, We obtain the solution of (30) as follows: where is an arbitrary constant, , and

For , let be the eigenvector of corresponding to , where , ; then expands to By Lemma 7, ; we put . By Theorem 1, , .

Denote , , and . By Lemma 6 we have It must be that Hence In the proof of Theorem 1, we have We complete the proof of Theorem 2.

Let be an eigenvalue of , and and are the algebraic multiplicity and the geometric multiplicity corresponding to , respectively.

If , by Theorem 1 and Theorem 2, and   , then is diagonalizable.

If , by Theorem 1, , , and   , where , .

By Theorem 2, , , and . We have the following:if is odd, then ;if is even, then , where , .

By Lemma 8 , Corollary 4 holds.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Natural Science Foundation of Guangdong Province, China (no. S2013010016994).