WestudytherelationshipbetweencertainGrobnerbasesforzero- dimensional radical ideals, and the vari... more WestudytherelationshipbetweencertainGrobnerbasesforzero- dimensional radical ideals, and the varieties dened by the ideals. Such a variety is a nite set of points in an ane n-dimensional space. We are in- terested in monomial orders that \eliminate" one variable, say z. Eliminating z corresponds to projecting points in n-space to (n 1)-space by discarding the z-coordinate. We show that knowing a minimal Grobner
We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and ... more We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine n-dimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z corresponds to projecting points in n-space to (n − 1)-space by discarding the z-coordinate. We show that knowing a minimal Gröbner basis under an elimination order immediately reveals some of the geometric structure of the corresponding variety, and knowing the variety makes available information concerning the basis. These relationships can be used to decompose polynomial systems into smaller systems.
Trends in Computational and Applied Mathematics, 2021
We present a linear-time algorithm that computes in a given real interval the number of eigenvalu... more We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.
Applicable Analysis and Discrete Mathematics, 2017
We present a linear time algorithm that computes the number of eigenvalues of a unicyclic graph i... more We present a linear time algorithm that computes the number of eigenvalues of a unicyclic graph in a given real interval. It operates directly on the graph, so that the matrix is not needed explicitly. The algorithm is applied to study the multiplicities of eigenvalues of closed caterpillars, obtain the spectrum of balanced closed caterpillars and give sufficient conditions for these graphs to be non-integral. We also use our method to study the distribution of eigenvalues of unicyclic graphs formed by adding a fixed number of copies of a path to each node in a cycle. We show that they are not integral graphs.
We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian ma... more We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian matrix of a tree in a given real interval. The algorithm can be applied to weightedor unweighted trees. Using our method we characterize the trees that have up to $5$ distincteigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacencyand normalized Laplacian matrices as special cases, among others.
WestudytherelationshipbetweencertainGrobnerbasesforzero- dimensional radical ideals, and the vari... more WestudytherelationshipbetweencertainGrobnerbasesforzero- dimensional radical ideals, and the varieties dened by the ideals. Such a variety is a nite set of points in an ane n-dimensional space. We are in- terested in monomial orders that \eliminate" one variable, say z. Eliminating z corresponds to projecting points in n-space to (n 1)-space by discarding the z-coordinate. We show that knowing a minimal Grobner
We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and ... more We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine n-dimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z corresponds to projecting points in n-space to (n − 1)-space by discarding the z-coordinate. We show that knowing a minimal Gröbner basis under an elimination order immediately reveals some of the geometric structure of the corresponding variety, and knowing the variety makes available information concerning the basis. These relationships can be used to decompose polynomial systems into smaller systems.
Trends in Computational and Applied Mathematics, 2021
We present a linear-time algorithm that computes in a given real interval the number of eigenvalu... more We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.
Applicable Analysis and Discrete Mathematics, 2017
We present a linear time algorithm that computes the number of eigenvalues of a unicyclic graph i... more We present a linear time algorithm that computes the number of eigenvalues of a unicyclic graph in a given real interval. It operates directly on the graph, so that the matrix is not needed explicitly. The algorithm is applied to study the multiplicities of eigenvalues of closed caterpillars, obtain the spectrum of balanced closed caterpillars and give sufficient conditions for these graphs to be non-integral. We also use our method to study the distribution of eigenvalues of unicyclic graphs formed by adding a fixed number of copies of a path to each node in a cycle. We show that they are not integral graphs.
We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian ma... more We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian matrix of a tree in a given real interval. The algorithm can be applied to weightedor unweighted trees. Using our method we characterize the trees that have up to $5$ distincteigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacencyand normalized Laplacian matrices as special cases, among others.
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Papers by Virgínia Maria Rodrigues