Enide Andrade Martins

Spectra of graphs obtained by a generalization of the join graph operation

Abstract Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

A generalization of Fiedler's lemma and some applications

In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145–156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedlers lemma is generalized and this generalization is applied to graph spectra and graph energy.

Computing the Laplacian spectra of some graphs

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path.

Eigenvalues of matrix commutators

We characterize the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size.

On the number of invariant polynomials of matrix commutators

We study the possible numbers of noneonstant invariant polynomials of the matrix commutator XA - AX, when X varies.

Spectra of weighted rooted graphs having prescribed subgraphs at some levels

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k j +1 (1 � jk). Let � � {1,2,...,k 1} and F= {Gj : j 2 �}, where Gj is a prescribed weighted graph on each set of children of B at the level k j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 +···+nk are characterized as the eigenvalues of symmetric tridiagonal matrices of order j, 1 � jk, easily constructed from the degrees of the vertices, the weights of the edges, and the eigenvalues of the matrices associated to the family of graphs F. These results are applied to characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph B (F) obtained from B and all the graphs in F = {Gj : j 2 �}; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.

On the eigenvalues of Jordan products

Abstract This paper studies the possible eigenvalues of the Jordan product XA+AX , when A is fixed and X varies.

Copies of a rooted weighted graph attached to an arbitrary weighted graph and applications

The spectrum of the Laplacian, signless Laplacian and adjacency matrices of the family of the weighted graphs R{H}, obtained from a connected weighted graph R on r vertices and r copies of a modified Bethe tree H by identifying the root of the i-th copy of H with the i-th vertex of R, is determined.

Feasibility Conditions on the Parameters of a Strongly Regular Graph

We consider a strongly regular graph, G, and associate a three dimensional Euclidean Jordan algebra, V, to the adjacency matrix A of G. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of V associated to A, we establish new feasibility conditions for the existence of strongly regular graphs.

On the invariant polynomials of Jordan products

Let F be an algebraically closed field. This paper describes the possible numbers of nonconstant invariant polynomials of the Jordan product XA+AX, when A is fixed and X varies.