The eigen-chromatic ratio of classes of graphs : molecular stability, asymptotes and area.

Master of Science in Mathematics, University of KwaZulu-Natal, Westville, 2017.This dissertation involves combining the two concepts of energy and the chromatic number of classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this ratio is the importance of its asymptotic convergence in applications, as well as the idea of area involving the Rieman integral of this ratio, when it is a function of the order n of the graph G belonging to a class of graphs. The energy of a graph G, is the sum of the absolute values of the eigenvalues associated with the adjacency matrix of G, and its importance has found its way into many areas of research in graph theory. The chromatic number of a graph G, is the least number of colours required to colour the vertices of the graph, so that no two adjacent vertices receive the same colour. The importance of ratios in graph theory is evident by the vast amount of research articles: Expanders, The central ratio of a graph, Eigen-pair ratio of classes of graphs , Independence and Hall ratios, Tree-cover ratio of graphs, Eigen-energy formation ratio, The eigen-complete difference ratio, The chromatic-cover ratio and "Graph theory and calculus: ratios of classes of graphs". We combine the two concepts of energy and chromatic number (which involves the order n of the graph G) in a ratio, called the eigen-chromatic ratio of a graph. The chromatic number associated with the molecular graph (the atoms are vertices and edges are bonds between the atoms) would involve the partitioning of the atoms into the smallest number of sets of like atoms so that like atoms are not bonded. This ratio would allow for the investigation of the effect of the energy on the atomic partition, when a large number of atoms are involved. The complete graph is associated with the value 1 2 when the eigen-chromatic ratio is investigated when a large number of atoms are involved; this has allowed for the investigation of molecular stability associated with the idea of hypo/hyper energetic graphs. Attaching the average degree to the Riemann integral of this ratio (as a function of n) would result in an area analogue for investigation. Once the ratio is defned the objective is to find the eigen-chromatic ratio of various well known classes of graphs such as the complete graph, bipartite graphs, star graphs with rays of length two, wheels, paths, cycles, dual star graphs, lollipop graphs and caterpillar graphs. Once the ratio of each class of graph are determined the asymptote and area of this ratio are determined and conclusions and conjectures inferred

A topological approach to neural complexity

Considerable efforts in modern statistical physics is devoted to the study of networked systems. One of the most important example of them is the brain, which creates and continuously develops complex networks of correlated dynamics. An important quantity which captures fundamental aspects of brain network organization is the neural complexity C(X)introduced by Tononi et al. This work addresses the dependence of this measure on the topological features of a network in the case of gaussian stationary process. Both anlytical and numerical results show that the degree of complexity has a clear and simple meaning from a topological point of view. Moreover the analytical result offers a straightforward algorithm to compute the complexity than the standard one.Comment: 6 pages, 4 figure