Small clique number graphs with three trivial critical ideals

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.Comment: 33 pages, 3 figure

Hardware Parallelization of Cores Accessing Memory with Irregular Access Patterns

This project studies FPGA-based heterogeneous computing architectures with the objective of discovering their ability to optimize the performances of algorithms characterized by irregular memory access patterns. The example used to achieve this is a graph algorithm known as Triad Census Algorithm, whose implementation has been developed and tested. First of all, the triad census algorithm is presented, explaining the possible variants and reviewing the existing implementations upon different architectures. The analysis focuses on the parallelization techniques which have allowed to boost performance, thus reducing execution time. Besides, the study tackles the OpenCL programming model, the standard used to develop the final application. Special attention is paid to the language details that have motivated some of the most important design decisions. The dissertation continues with the description of the project implementation, including the application objectives, the system design, and the different variants developed to enhance algorithm performance. Finally, some of the experimental results are presented and discussed. All implemented versions are evaluated and compared to decide which is the best in terms of scalability and execution time