Eliseu Fritscher

CHARACTERIZING TREES WITH LARGE LAPLACIAN ENERGY

We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer n, we find a class of cardinality approximately p n whose elements are the n-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum Sk(T) of the k largest Laplacian eigenvalues of an n-vertex tree T with diameter at least four, where k 2 f1;:::;ng.

Exploring Symmetries to Decompose Matrices and Graphs Preserving the Spectrum

Given a special kind of symmetric matrix, we present a decomposition technique that preserves its spectrum. We transform this result into an algorithm for graphs. The algorithm disconnects the graph, resulting in smaller matrices, reducing the complexity of computing the graph spectrum. This technique may be seen as a unified approach of several decomposition techniques present in the literature for different instances of matrices and classes of graphs. As an application, we use the algorithm for three classes of graphs: threshold graphs, generalized Bethe trees, and multifan graphs, obtaining expressions for their spectra, for various matrices.

Loopy, Hankel, and combinatorially skew-Hankel tournaments

We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type.

Unicyclic graphs with equal Laplacian energy

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of noncospectral unicyclic graphs on vertices with the same Laplacian energy.

Tournaments associated with multigraphs and a theorem of Hakimi

A tournament of order n is usually considered as an orientation of the complete graph K n . In this note, we consider a more general definition of a tournament that we call a C -tournament, where C is the adjacency matrix of a multigraph G , and a C -tournament is an orientation of G . The score vector of a C -tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C -tournament with a prescribed score vector R and gave an algorithm to construct such a C -tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimis theorem, and then provide a direct construction of such a C -tournament which works even for weighted graphs.

Trace index of a (0,1)-matrix

For -matrices in general, and for -matrices with a constant number of 1s in each row and column, we determine the possible sets of traces of such matrices as its rows are permuted. Except for some special matrices, which are identified, the set of such traces is a consecutive set of integers from .