Oscar Rojo

An always nontrivial upper bound for Laplacian graph eigenvalues

Abstract Let G be a graph on vertex set V= v 1 ,v 2 ,…,v n . Let d i be the degree of v i , let N i be the set of neighbours of v i and let |S| be the number of vertices of S⊆V. In this note, we prove that max d i +d j −|N i ∩N j |:1⩽i is an upper bound for the largest eigenvalue of the Laplacian matrix of G. For any G, this bound does not exceed the order of G.

Symmetric nonnegative realization of spectra

A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado’s extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.

Existence and construction of nonnegative matrices with complex spectrum

Abstract The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ ={ λ 1 , λ 2 ,…, λ n }, find necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ . Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0 ⩾max 2⩽ j ⩽ n | λ j | such that the problem has a solution if and only if λ 1 ⩾ λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.

The spectrum of the Laplacian matrix of a balanced binary tree

Abstract Let L ( B k ) be the Laplacian matrix of an unweighted balanced binary tree B k of k levels. We prove that spectrum of L ( B k ) is σ L ( B k ) =⋃ j=1 k−1 σ( T j )∪σ( S k ), where, for 1⩽j⩽k−1, T j is the j × j principal submatrix of the tridiagonal k × k matrix S k , S k = 1 2 0 ⋯ 0 2 3 2 ⋱ ⋮ 0 2 ⋱ ⋱ 0 ⋮ ⋱ ⋱ 3 2 0 ⋯ 0 2 2 . We derive that the multiplicity of each eigenvalue of T j ,1⩽j⩽k−1, as an eigenvalue of L ( B k ), is at least 2 k − j −1 . Finally, for each T j , using some results in [Electron. J. Linear Algebra 6 (2000) 62], we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue. In particular, we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of L ( B k ) .

Bounds for sums of eigenvalues and applications

Let A be a matrix of order n × n with real spectrum λ1 ≥ λ2 ≥ ⋯ ≥ λn. Let 1 ≤ k ≤ n − 2. If λn or λ1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenvalues of A. Then, we obtain a majorization vector for (λ1, λ2,…, λn−1) when λn is known and a majorization vector for (λ2, λ3,…, λn) when λ1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum.

New eigenvalue estimates for complex matrices

Abstract In a recent work [1], we have proved that all the eigenvalues λj of a complex matrix A of order n lie in a disk with center at tr A n and radius [ n−1 n (Σ k=1 n |λ k | 2 − | tr A | 2 n )] 1 2 . In this paper, we prove that the eigenvalues of A are contained in a rectangle with vertices on the boundary circle of the mentioned disk, improving in this way the result in [1]. We recall some inequalities which alloww us to bound the radius of the disk and the rectangle. The bounds can be computed without knowing the eigenvalues of A.

A Decreasing Sequence of Eigenvalue Localization Regions

Abstract We construct a decreasing sequence of rectangles (Rp) in such a way that all the eigenvalues of a complex matrix A are contained in each rectangle. When A is a matrix with real spectrum or a normal matrix, each Rp can be obtained without knowing the eigenvalues of A.

On nonnegative realization of partitioned spectra

We consider partitioned lists of real numbers � = {�1,�2,...,�n}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum �. Our results extend the ones given in (R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844- 856, 2006.) and (R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1-18, 2007.) for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries.

EXTREMAL ALGEBRAIC CONNECTIVITIES OF CERTAIN CATERPILLAR CLASSES AND SYMMETRIC CATERPILLARS

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd 1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = (p1,p2,...,pd 1) such that p1 ≥ 1,p2 ≥ 1,...,pd 1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1,Sp2,...,Spd 1 and the path Pd 1 by identifying the root of Spi with the i−vertex of Pd 1. Let n > 2(d − 1) be given. Let In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a caterpillar in S as the smallest eigenvalue of a 2 × 2 - block tridiagonal matrix of order 2s × 2s if d = 2s + 1 or d = 2s + 2 is characterized.

Spectra of weighted compound graphs of generalized Bethe trees

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let {Bi :1≤ i ≤ m} be a set of trees such that, for i =1 ,2,...,m , (i) Bi is a generalized Bethe tree of ki levels, (ii) the vertices ofBi at the level j have degree di,ki−j+1 for j =1 ,2,...,ki, and (iii) the edges ofBi joining the vertices at the level j with the vertices at the level (j + 1) have weight wi,ki−j for j =1 ,2,...,k i −1. LetGm{Bi :1≤ i ≤ m}be the graph obtained fromGm and the trees B1,B2,...,Bm by identifying the root vertex of Bi with the ith vertex of Gm. A complete characterization is given of the eigenvalues of the Laplacian and adjacency matrices ofGm {Bi :1≤ i ≤ m} together with results about their multiplicities. Finally, these results are applied to the particular case B1 = B2 = ···= Bm.