In combinatorial mathematics, a Steiner system is a type of block design. A Steiner triple system is a special case of Steiner system where all blocks contain 3 elements and each pair of points occurs in exactly one block. Independent sets in Steiner triple systems is the topic which is discussed in this thesis. Some properties related to independent sets in Steiner triple system are provided. The distribution of sizes of maximum independent sets of Steiner triple systems of specific order is also discussed in this thesis. An algorithm for constructing a Steiner triple system with maximum independent set whose size is restricted with a lower bound is provided. An alternative way to construct a Steiner triple system using an affine plane is also presented. A modified greedy algorithm for finding a maximal independent set in a Steiner triple system and a post-optimization method for improving the results yielded by this algorithm are established.

Quantum computers provide a promising future, where computationally difficult
problems can be executed exponentially faster than the current classical computers we have in use today. While there is tremendous research and development in the creation of quantum computers, there is a fundamental challenge that exists in the quantum world. Due to the fragility of the quantum world, error correction methods have originated since 1995 to tackle the giant problem. Since the birth of the idea that these powerful computers can crunch and process numbers beyond the limit of the current computers, there exist several mathematical error correcting codes that could potentially give the required stability in the fragile and fault tolerant quantum world. While there has been a multitude of possible solutions, there is no one single error correcting code that is the key to solving the problem. Almost every solution presented has shared with it a limiting factor or an issue that prevents it from becoming the breakthrough that is desperately needed.

This paper gives an introductory knowledge of what is the quantum world and why there is a need for error correcting topologies. Finally, it introduces one recent topology that could be added to the list of possible solutions to this central problem. Rather than focusing on the mathematical frameworks, the paper introduces the main concepts so that most readers even outside the major field of computer science can understand what the main problem is and how this topology attempts to solve it.

When designing screening experiments for many factors, two problems quickly arise. The first is that testing all the different combinations of the factors and interactions creates an experiment that is too large to conduct in a practical amount of time. One way this problem is solved is with a combinatorial design called a locating array (LA) which can efficiently identify the factors and interactions most influential on a response. The second problem is how to ensure that combinations that prohibit some particular tests are absent, a requirement that is common in real-world systems. This research proposes a solution to the second problem using constraint satisfaction.

Error-correcting codes are fundamental in modern digital communication with applications in data storage and data transmission. Interest in a class of error-correcting codes called low-density parity-check (LDPC) codes has been growing since their recent rediscovery because of their low decoding complexity and their high-performance. However, practical applications have been limited due to the difficulty of finding good LDPC codes for practical parameters. This paper proposes an exhaustive and a randomized algorithm for constructing a family of LDPC codes with practical parameters whose matrix representations meet the following requirements: for each row in the LDPC code matrix there exists exactly one common nonzero element, each row has a minimum weight of one and must be odd, and each column has a weight of at least two. These conditions improve performance of the resulting codes and simplify conversion into codes for quantum systems. Both algorithms utilize a maximal clique algorithm to construct LDPC matrices from graphs whose vertices are possible rows within said matrices and are adjacent the first condition is true. While these algorithms were found to be suitable for small parameters, future work which optimizes the resulting codes for their expected applications could also dramatically increase performance of the algorithms themselves.

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.

Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.

In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.

Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.

Medium access control (MAC) is a fundamental problem in wireless networks.

In ad-hoc wireless networks especially, many of the performance and scaling issues

these networks face can be attributed to their use of the core IEEE 802.11 MAC

protocol: distributed coordination function (DCF). Smoothed Airtime Linear Tuning

(SALT) is a new contention window tuning algorithm proposed to address some of the

deficiencies of DCF in 802.11 ad-hoc networks. SALT works alongside a new user level

and optimized implementation of REACT, a distributed resource allocation protocol,

to ensure that each node secures the amount of airtime allocated to it by REACT.

The algorithm accomplishes that by tuning the contention window size parameter

that is part of the 802.11 backoff process. SALT converges more tightly on airtime

allocations than a contention window tuning algorithm from previous work and this

increases fairness in transmission opportunities and reduces jitter more than either

802.11 DCF or the other tuning algorithm. REACT and SALT were also extended

to the multi-hop flow scenario with the introduction of a new airtime reservation

algorithm. With a reservation in place multi-hop TCP throughput actually increased

when running SALT and REACT as compared to 802.11 DCF, and the combination of

protocols still managed to maintain its fairness and jitter advantages. All experiments

were performed on a wireless testbed, not in simulation.

Polar ice masses can be valuable indicators of trends in global climate. In an effort to better understand the dynamics of Arctic ice, this project analyzes sea ice concentration anomaly data collected over gridded regions (cells) and builds graphs based upon high correlations between cells. These graphs offer the opportunity to use metrics such as clustering coefficients and connected components to isolate representative trends in ice masses. Based upon this analysis, the structure of sea ice graphs differs at a statistically significant level from random graphs, and several regions show erratically decreasing trends in sea ice concentration.

Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of permutations required to cover each subsequence, and in finding an explicit construction of such a set of permutations that has size close to or equal to the minimum possible. The construction of such permutation coverings has proven to be computationally difficult. While many examples for permutations of small length have been found, and strong asymptotic behavior is known, there are few explicit constructions for permutations of intermediate lengths. Most of these are generated from scratch using greedy algorithms. We explore a different approach here. Starting with a set of permutations with the desired coverage properties, we compute local changes to individual permutations that retain the total coverage of the set. By choosing these local changes so as to make one permutation less "essential" in maintaining the coverage of the set, our method attempts to make a permutation completely non-essential, so it can be removed without sacrificing total coverage. We develop a post-optimization method to do this and present results on sequence covering arrays and other types of permutation covering problems demonstrating that it is surprisingly effective.

As an example of "big data," we consider a repository of Arctic sea ice concentration data collected from satellites over the years 1979-2005. The data is represented by a graph, where vertices correspond to measurement points, and an edge is inserted between two vertices if the Pearson correlation coefficient between them exceeds a threshold. We investigate new questions about the structure of the graph related to betweenness, closeness centrality, vertex degrees, and characteristic path length. We also investigate whether an offset of weeks and years in graph generation results in a cosine similarity value that differs significantly from expected values. Finally, we relate the computational results to trends in Arctic ice.

The Tamari lattice T(n) was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Since then it has been studied in various areas of mathematics including cluster algebras, discrete geometry, algebraic combinatorics, and Catalan theory. Although in several related lattices the number of maximal chains is known, the enumeration of these chains in Tamari lattices is still an open problem.

This dissertation defines a partially-ordered set on equivalence classes of certain saturated chains of T(n) called the Tamari Block poset, TB(lambda). It further proves TB(lambda) is a graded lattice. It then shows for lambda = (n-1,...,2,1) TB(lambda) is anti-isomorphic to the Higher Stasheff-Tamari orders in dimension 3 on n+2 elements. It also investigates enumeration questions involving TB(lambda), and proves other structural results along the way.