Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

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.

Development of a single photon counting computed tomography system using MPGDs

The development of computed tomography systems with energy resolving detectors is a current challenge in medical physics and biomedical engineering. A computed tomography system of this kind allows getting complementary informations relatively to conventional systems, that can help the medical diagnosis, being of great interest in medicine. The work described in this thesis is related to the development of a computed tomography system using micropattern gaseous detectors, which allow storing, simultaneously, information about the interaction position and the energy of each single photon that interacts with the detector. This kind of detectors has other advantages concerning the cost and characteristics of operation when compared with solid state detectors. Tomographic acquisitions were performed using a MicroHole & Strip Plate based detector, which allowed reconstructing cross-sectional images using energy windows, applying the energy weighting technique and performing multi-slice and tri-dimensional reconstructions. The contrast-to-noise ratio was improved by 31% by applying the energy weighting technique, comparing with the corresponding image obtained with the current medical systems. A prototype of a computed tomography with flexibility to change the detector was developed, making it possible to apply different detectors based on Thick-COBRA. Several images acquired with these detectors are presented and demonstrate their applicability in X-ray imaging. When operating in NeCH4, the detector allowed a charge gain of 8 104, an energy resolution of 20% (full width at half maximum at 8 keV), a count rate of 1 106 Hz/mm2, a very stable operation (gain fluctuations below 5%) and a spacial resolution of 1.2 mm for an energy photon of 3.6 keV. Operating the detector in pure Kr allowed increasing the detection efficiency and achieving a charge gain of 2 104, an energy resolution of 32% (full width at half maximum at 22 keV), a count rate of 1 105 Hz/mm2, very stable operation and a spatial resolution of 500 m. The software already existing in the group was improved and tools to correct geometric misalignments of the system were also developed. The reconstructions obtained after geometrical correction are free of artefacts due to the referred misalignments.