Applications of Graph Theory and System Theory to Large Scale Systems

In graph theory, the minimum cardinality infimum measure of set colorings, phasings, and intersection assignments has been studied for the important cases where there is no restriction on the sets, where each set is a real interval, and where each set is a consecutive set of integers. The maximum cardinality score supremum measure score has also been studied in these cases. Progress has included the development of general procedures, explicit formulas, and efficient algorithms. Related work has explored a series of ultimate numbers related to the n-chromatic numbers, and the structural characterization of certain classes of graphs of boxicity at most 2, which generalize the interval graphs. In system theory, a computationally oriented approach to nonlinear system regulation has been developed, based on the notion of piecewise-linear systems. The necessary algebraic concepts had to be themselves developed during the course of the research. New designs and theoretical results were obtained also for the control and observation of parametrized families of systems, and for delay and other well-known types of systems.