Improving bounds on large instances of graph coloring

This thesis explores new methods, using both vertex cover and exact graph coloring algorithms in addition to our implementation of the state of the art, to develop a hybrid algorithm that on most instances is able to tighten the bounds or determine the optimal number of colors outright

Efficient enumeration of small graphlets and orbits

As the world is flooded with data, the demand for mining data for useful purposes is increasing. An effective techniques is to model the data as networks (graphs) and then apply graph mining techniques for analysis. As on date, the algorithms available to count graphlets and orbits for various types of graphs and their generalizations are limited. The thesis aims to fill the gap by presenting a simple and efficient algorithm for 3-node graphlet and orbit counting that is generic enough to work for both undirected and directed graphs. Our algorithm is compared with the state-of-art algorithms and we show that in most cases our algorithm performs better. We demonstrate our algorithm in three case studies related to (i) enzyme and metabolite correlation network in corn, (ii) watershed governance networks, and (iii) patterns exhibited by co-expression networks of healthy and cancerous stomach cells

Coloring and degeneracy for determining very large and sparse derivative matrices

Estimation of large sparse Jacobian matrix is a prerequisite for many scientific and engineering problems. It is known that determining the nonzero entries of a sparse matrix can be modeled as a graph coloring problem. To find out the optimal partitioning, we have proposed a new algorithm that combines existing exact and heuristic algorithms. We have introduced degeneracy and maximum k-core for sparse matrices to solve the problem in stages. Our combined approach produce better results in terms of partitioning than DSJM and for some test instances, we report optimal partitioning for the first time