On the Faria's inequality for the Laplacian and signless Laplacian spectra: a unified approach

Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.

Ensaios sobre o crescimento económico regional e a dinâmica do processo de convergência

The research is centred on economic growth and it adopts a regional view focused on European regions (NUTS 2). Under analysis is the undergoing convergence process in Europe to evaluate if there is an increase in regional economic growth and a simultaneous decrease in regional disparities. The empirical evidence shows signs of transience on the regional convergence process, along with the formation of several clusters of regions. Considering this evidence it is of interest to assess the profiles of such clusters, and evaluate the homogeneousness of its regions and their spread across European space. As suggested in the literature, space is a relevant factor in the spread of clusters, according to the different variables (regions profiles, diversity measures and growth indicators). Regional disparities across Europe are, on average, large and persistent, with a high degree of stability among the relative rankings of European regions. Such results cast shadows about the impact of European structural funding in diminishing regional disparities. Such an approach is of interest due to the growing need to evaluate the effectiveness and efficiency of public policies implemented. The fact that there is an undergoing process of economic integration in Europe increases the usefulness of such an analysis, especially in a context in which European public funds are negotiated and granted.