Israel Rocha

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.

Bounding the sum of the largest Laplacian eigenvalues of graphs

We prove that Brouwers conjecture holds for certain classes of graphs. We also give upper bounds for the sum of the largest Laplacian eigenvalues for graphs satisfying certain properties: those that contain a path or a cycle of a given size, graphs with a given matching number and graphs with a given maximum degree. Then we provide conditions for which these upper bounds are better than the previous known results.

Absolute algebraic connectivity of double brooms and trees

We use a geometric technique based on embeddings of graphs to provide an explicit formula for the absolute algebraic connectivity and its eigenvectors of double brooms. Besides, we give a polynomial time combinatorial algorithm that computes the absolute algebraic connectivity of a given tree.

Algebraic connectivity on a subclass of caterpillars

Abstract We study a subfamily - which we call A q - of the family of trees known as caterpillars. We show that all but one tree in A q is a type II tree. We give bounds for the algebraic connectivity in A q and exhibit the tree attaining the bounds. Finally we give a total order in A q by algebraic connectivity.

Spectral Bisection with Two Eigenvectors

Abstract We show a spectral bisection algorithm which makes use of the second and third eigenvector of the Laplacian matrix. This algorithm is guaranteed to return a cut that is smaller or equal to the one returned by the classic spectral bisection. To this end, we investigate combinatorial properties of certain configurations of a graph partition. These properties, that we call organized partitions, are shown to be related to the minimality and maximality of a cut. We show that organized partitions are related to the third eigenvector of the Laplacian matrix and give bounds on the minimum cut in terms of organized partitions and eigenvalues.