Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.     

A look-ahead Bareiss algorithm for general Toeplitz matrices

Summary. The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order~ with only~ arithmetic operations, instead of~ operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems. Received August 27, 1993 / Revised version received March 1994     

Interlacing eigenvalues on some operations of graphs

In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing.     

A NOTE ON MERRIS＇ CONJECTURES

The Merris‘ conjectures for threshold graphs and d-regular graphs are proved. Andthe one of the conjectures of Merris holds for trees is also proved.     

New proofs for Levine’s theorems

Recently, Levine [9] expressed the vertex weighted complexity on spanning trees (with a fixed root) of the directed line graph of a digraph D in terms of the edge weighted complexity on spanning trees (with a fixed root) of D. We present new proofs for two Levine’s Theorems. Furthermore, we express the number of unicycles of the directed line graph of a digraph D in terms of the number of unicycles of D.     

Determinant identities for Laplace matrices

We show that every minor of an n×n Laplace matrix, i.e., a symmetric matrix whose row- and column sums are 0, can be written in terms of those minors that are obtained by deleting two rows and the corresponding columns. The proof is based on a classical determinant identity due to Sylvester. Furthermore, we show how our result can be applied in the context of electrical networks and spanning tree enumeration.     

Polynomial perturbations of bilinear functionals and Hessenberg matrices

This paper deals with symmetric and non-symmetric polynomial perturbations of symmetric quasi-definite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals.     

Noisy random graphs and their Laplacians

Spectra and representations of some special weighted graphs are investigated with weight matrices consisting of homogeneous blocks. It is proved that a random perturbation of the weight matrix or that of the weighted Laplacian with a “Wigner-noise” will not have an effect on the order of the protruding eigenvalues and the representatives of the vertices will unveil the underlying block-structure.Such random graphs adequately describe some biological and social networks, the vertices of which belong either to loosely connected strata or to clusters with homogeneous edge-densities between any two of them, like the structure guaranteed by the Regularity Lemma of Szemerédi.     

Eigenvalues and degree deviation in graphs

Let G be a graph with n vertices and m edges and let μ(G) = μ₁(G) ? ? ? μ_n(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑_u∈V(G)∣d(u)-2m/n∣. We prove that

     

Fast algorithms for Toeplitz and Hankel matrices

The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

Characteristic polynomial of a generalized complete product of matrices

For a simple graph G, let denote the complement of G relative to the complete graph and let P_G(x)=det(xI-A(G)) where A(G) denotes the adjacency matrix of G. The complete product G∇H of two simple graphs G and H is the graph obtained from G and H by joining every vertex of G to every vertex of H. In [2]P_G∇H(x) is represented in terms of P_G, , P_H and . In this paper we extend the notion of complete product of simple graphs to that of generalized complete product of matrices and obtain their characteristic polynomials.     

Even and odd diagonals in doubly stochastic matrices

The existence of even or odd diagonals in doubly stochastic matrices depends on the number of positive elements in the matrix. The optimal general lower bound in order to guarantee the existence of such diagonals is determined, as well as their minimal number for given number of positive elements. The results are related to the characterization of even doubly stochastic matrices in connection with Birkhoff's algorithm.     

Projective lines over Jordan systems and geometry of Hermitian matrices

Any set of σ-Hermitian matrices of size n×n over a field with involution σ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

On the sum of Laplacian eigenvalues of graphs

Let k be a natural number and let G be a graph with at least k vertices. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most , where e(G) is the number of edges of G. We prove this conjecture for k=2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k-1.     

The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (, , P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX _pq with the interpretation thatX _pq () is the distance betweenp andq at the instant . The probability of the eventJ _pqr = { :X _pr ()X _pq ()+X _qr ()} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, Statistical Metrics, Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, Statistical Metric Spaces, Pacific J. Math.10 (1960), 313–334). It turns out that for > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X < Y + Z) < . This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J _pqr ) = 1.     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.