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The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited in the now classical paper by Golub and Welsch in 1969. From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been reduced to solving direct and inverse eigenvalue problems… (More)

In this paper we study both direct and inverse eigenvalue problems for diagonal-plus-semiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we devise a set… (More)

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the max-imization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix M , defined in terms of the… (More)

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on… (More)

Nodal theorems for generalized modularity matrices ensure that the cluster located by the positive entries of the leading eigenvector of various mod-ularity matrices induces a connected subgraph. In this paper we obtain lower bounds for the modularity of that set of nodes showing that, under certain conditions, the nodal domains induced by eigenvectors… (More)