Graph Laplacians, Nodal Domains, and Hyperplane Arrangements

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e., the connected components of the maximal induced subgraphs of G on which an eigenvector # does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of # in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures

Fuel Cell Technologies, Applications, and State of the Art. A Reference Guide

The reliance upon fossil fuels is one of the most important problems that we have to deal with nowadays, because using them is highly not sustainable, and leads to serious environmental issues, such as air pollution and global warming, beside its negative effect on the economic security and development. Alternatively, environmental-friendly, sustainable, efficient new power sources are needed. Among all of the different technologies associated with the renewable energy, fuel cell technologies represent one of the most promising ones. In this article, an overview of the technology and its advantages and disadvantages compared with competitive technologies was revealed. Application of different types of fuel cells is covered in the stationary, portable and transport power sectors. Furthermore, a special focus was made on the proton exchange membrane (PEM) fuel cells; in that its structure was studied in detail. The current limitation and promising development in PEM technology were stated