Vilmar Trevisan

Polynomial Factorization

A new coefficient bound is established for factoring univariate polynomials over the integers. Unlike an overall bound, the new bound limits the size of the coefficients of at least one irreducible factor of the given polynomial. The single-factor bound is derived from the weighted norm introduced in Beauzamy et al. (1990) and is almost optimal. Effective use of this bound in p-adic lifting results in a more efficient factorization algorithm. A full example and comparisons with known coefficient bounds are included.

Laplacian energy of diameter 3 trees

Abstract Let T n be the set of all trees of diameter 3 and n vertices. We show that the Laplacian energy of any tree in T n is strictly between the Laplacian energy of the path P n and the star S n , partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in T n . Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in T n .

Distance- k knowledge in self-stabilizing algorithms

Many graph problems seem to require knowledge that extends beyond the immediate neighbors of a node. The usual self-stabilizing model only allows for nodes to make decisions based on the states of their immediate neighbors. We provide a general transformation for constructing self-stabilizing algorithms which utilize distance-k knowledge. Our transformation has both a slowdown and space overhead in n^O^(^l^o^g^k^), and might be thought of as a distance-k resource allocation algorithm. Our main application is a polynomial-time self-stabilizing algorithm for finding maximal irredundant sets, a problem which seems to require distance-4 information. These results can be generalized to efficiently find maximal P-sets, for properties P which we call local monotonic. Our techniques extend results in a recent paper by Gairing et al. for achieving distance-two information.

Reducing the adjacency matrix of a tree

Let T be a tree, A its adjacency matrix, and a scalar. We describe a linear-time algorithm for reducing the matrix In +A. Applications include computing the rank of A, nding a maximum matching in T , computing the rank and determinant of the associated neighborhood matrix, and computing the characteristic polynomial of A.

Distance- k information in self-stabilizing algorithms

Many graph problems seem to require knowledge that extends beyond the immediate neighbors of a node. The usual self-stabilizing model only allows for nodes to make decisions based on the states of their immediate neighbors. We provide a general polynomial transformation for constructing self-stabilizing algorithms which utilize distance-shape k knowledge, with a slowdown of nO(log k). Our main application is a polynomial-time self-stabilizing algorithm for finding maximal irredundant sets, a problem which seems to require distance-4 information. We also show how to find maximal k-packings in polynomial-time. Our techniques extend results in a recent paper by Gairing et al. for achieving distance-two information

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.

An O(n2) Algorithm for the Characteristic Polynomial of a Tree

Abstract Abstract We describe an O(n2) algorithm to find the characteristic polynomial of the adjacency matrix of any tree.

The determinant of a tree's neighborhood matrix

Abstract Let N be an n × n neighborhood matrix for a tree or forest. We show that ¦ det N¦ is bounded by the nth Fibonacci number. We obtain a simple, elegant algorithm to compute det N that operates directly on the forest and uses O(n) space and O(n) arithmetic operations.

Threshold graphs of maximal Laplacian energy

The Laplacian energy of a graph is defined as the sum of the absolute values of the differences of average degree and eigenvalues of the Laplacian matrix of the graph. This spectral graph parameter is upper bounded by the energy obtained when replacing the eigenvalues with the conjugate degree sequence of the graph, in which the i th number counts the nodes having degree at least i . Because the sequences of eigenvalues and conjugate degrees coincide for the class of threshold graphs, these are considered likely candidates for maximizing the Laplacian energy over all graphs with given number of nodes. We do not answer this open problem, but within the class of threshold graphs we give an explicit and constructive description of threshold graphs maximizing this spectral graph parameter for a given number of nodes, for given numbers of nodes and edges, and for given numbers of nodes, edges and trace of the conjugate degree sequence in the general as well as in the connected case. In particular this positively answers the conjecture that the pineapple maximizes the Laplacian energy over all connected threshold graphs with given number of nodes.

ON THE DISTRIBUTION OF LAPLACIAN EIGENVALUES OF TREES

Abstract For a tree T with n vertices, we apply an algorithm due to Jacobs and Trevisan (2011) to study how the number of small Laplacian eigenvalues behaves when the tree is transformed by a transformation defined by Mohar (2007). This allows us to obtain a new bound for the number of eigenvalues that are smaller than 2. We also report our progress towards a conjecture on the number of eigenvalues that are smaller than the average degree.