Kirchhoff index of a non-complete wheel

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 LaplacianPostprint (author's final draft

The group inverse of extended symmetric and periodic Jacobi matrices

In this work, we explicitly compute the group inverse of symmetric and periodic Jacobi matrices with constant elements that have been extended by adding a row and a column conveniently de ned. For this purpose, we interpret such matrices as the combinatorial Laplacian of a non-complete wheel that has been obtained by adding a vertex to a cycle and some edges conveniently chosen. The obtained group inverse is an incomplete block matrix with a block Toeplitz structure. In addition, we obtain the e ffective resistances and the Kirchhoff index of non-complete wheels.Preprin

Matriz Laplaciana de grafos ponderados con vértices independientes

Postprint (author's final draft

The betweenness centrality of a graph

A measure of the centrality of a vertex of a graph is the portion of shortest paths crossing through it between other vertices of the graph. This is called betweenness centrality and here we study some of its general properties, relations with distance parameters (diameter, mean distance), local parameters, symmetries, etc. Some bounds for this parameter are obtained, using them to improve known bounds for the mean distance of the graph

Green matrices of weighted graphs with pendant vertices

Postprint (published version

Eigenvalue distribution in scale free graphs

Scale free graphs can be found very often as models of real networks and are characterized by a power law degree distribution, that is, for a constant $\gamma\geq 1$ the number of vertices of degree $d$ is proportional to $d^{-\gamma}$. Experimental studies show that the eigenvalue distribution also follows a power law for the highest eigenvalues. Hence it has been conjectured that the power law of the degrees determines the power law of the eigenvalues. In this paper we show that we can construct a scale free graph with non highest eigenvalue power law distribution. For $\gamma=1$ we can construct a scale free graph with small spectrum and a regular graph with eigenvalue power law distribution

Bounded expansion in models of webgraphs

We study the bounded expansion of several models of web graphs. We show that various deterministic graph models for large complex networks have constant bounded expansion.We study two random models of webgraphs, showing that the model of Bonato has not bounded expansion, and we conjecture that the classical model of Barabási may have also not bounded expansion

On golden spectral graphs

The concept of golden spectral graphs is introduced and some of their general properties reported. Golden spectral graphs are those having a golden proportion for the spectral ratios defined on the basis of the spectral gap, spectral spread and the difference between the second largest and the smallest eigenvalue of the adjacency matrix. They are good expanders and display excellent synchronizability. Here we report some new construction methods as well as several of their topological parameters

Betweenness-selfcentric graphs

The betweenness centrality of a vertex of a graph is the portion of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality.Preprin

Notes on betweenness centrality of a graph

The betweenness centrality of a vertex of a graph is the portion of shortest paths between all pairs of vertices passing through that vertex. We study selected general properties of this invariant and its relations to distance parameters (diameter, mean distance); also, there are studied properties of graphs whose vertices have the same value of betweenness centrality