Silvia Gago

Notes on the betweenness centrality of a graph

The betweenness centrality of a vertex of a graph is the portion of the shortest paths between all pairs of vertices passing through a given vertex. We study upper bounds for this invariant and its relations to the diameter and average distance of a graph.

Synchronizability of complex networks

Recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems, and in particular their synchronization. In this paper, by using known and new results from spectral graph theory, we characterize relevant factors which affect the synchronization of complex networks.

A star-based model for the eigenvalue power law of Internet graphs

Using a simple deterministic model for the Internet graph we show that the eigenvalue power-law distribution for its adjacency matrix is a direct consequence of the degree distribution and that the graph must contain many star subgraphs.

The inverses of some circulant matrices

We present here necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three parameters and moreover, we explicitly compute the inverse. Our study also encompasses a wide class of circulant symmetric matrices. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric, circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.

Laplacian matrix of a weighted graph with new pendant vertices

Abstract The Laplacian matrix of a simple graph has been widely studied, as a consequence of its applications. However the Laplacian matrix of a weighted graph is still a challenge. In this work we provide the Moore-Penrose inverse of the Laplacian matrix of the graph obtained adding new pendant vertices to an initial graph, in terms of the Moore-Penrose inverse of the Laplacian matrix of the original graph. As an application we can compute the effective resistances and the Kirchhoff index of the new network.

Jacobi matices and boundary value problems in distance-regular graphs

Regular boundary value problems on a distance-regular graph associated with Schrodinger operators are analyzed. These problems include the cases in which the boundary has one or two vertices. In each case, the Green matrices are given in terms of two families of orthog- onal polynomials, one of them corresponding with the distance polynomials of the distance-regular graphs.

Kirchhoff index of the connections of two networks by an edge

Abstract In this work we compute the group inverse of the Laplacian of the connections of two networks by and edge in terms of the Laplacians of the original networks. Thus the effective resistances and Kirchhoff index of the new network can be derived from the Kirchhoff indexes of the original networks.

Kirchhoff index of a non-complete wheel

Abstract In this work, we compute analitycally the Kirchhoff index and effective resistances of a weighted non–complete wheel that has been obtained by adding a vertex to a weighted cycle and some edges conveniently chosen. To this purpose we use the group inverse of the combinatorial Laplacian.

Boundary Value Problems for Schrödinger Operators on a Path Associated to Orthogonal Polynomials

In this work, we concentrate on determining explicit expressions, via suitable orthogonal polynomials on the line, for the Green function associated with any regular boundary value problem on a weighted path, whose weights are determined by the coefficients of the three-term recurrence relation.