Analyzes the results of M.I. Ostrovskii's theorem of inequalities which estimate the minimal edge congestion for finite simple graphs. Uses the generic results of the theorem to examine and further reduce the parameters of inequalities for specific families of graphs, particularly complete graphs and complete bipartite graphs. Also, explores a possible minimal congestion tree for some grids while forming a conjecture for all grids.

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Chapter I begins with a brief history of the topic of greatest common subgraphs. Then we provide a summaiy of the work done on some variations of greatest common subgraphs. Finally, in this chapter we present results previously obtained on greatest common divisors and least common multiples of graphs. In Chapter II the concepts of prime graphs, prime divisors of graphs, and primeconnected graphs are presented. We show the existence of prime trees of any odd size and the existence of prime-connected trees that are not prime having any odd composite size. Then the number of prime divisors in a graph is studied. Finally, we present several results involving the existence of graphs whose size satisfies some prescribed condition and which contains a specified number of prime divisors. Chapter III presents properties of greatest common divisors and least common multiples of graphs. Then graphs with a prescribed number of greatest common divisors or least common multiples are studied. In Chapter IV we study the sizes of greatest common divisors and least common multiples of specified graphs. We find the sizes of greatest common divisors and least common multiples of stars and that of stripes. Then the size of greatest common divisors and least common multiples of paths and complete graphs are investigated. In particular, the size of least common multiples of paths versus K3 or K4 are determined. Then we present the greatest common divisor index of a graph and we determine this parameter for several classes of graphs. iii In Chapter V greatest common divisors and least common multiples of digraphs are introduced. The existence of least common mutliples of two stars is established, and the size of a least common multiple is found for several pairs of stars. Finally, we present the concept of greatest common divisor index of a digraph and determine it for several classes of digraphs. iv / Mathematical Sciences / Ph. D. (Mathematical sciences)

published_or_final_version / Electrical Engineering / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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A strongly regular graph can be design partitioned if the vertices of the graph can be partitioned into two sets V and B such that V is a coclique and every vertex in B is adjacent to the same number of vertices in V. In this case, a balanced incomplete block design can be defined by taking elements of V as objects and elements of B as blocks. Many strongly regular graphs can be design partitioned. The nation of design partitioning is extended to a partitioning by a generalization of block designs called order-free designs. All strongly regular graphs can be partitioned via order-free designs. Order-free designs are used to show the nonexistence of a strongly regular graph with parameters (50,28,18,12). The existence of this graph was previously undecided. A computer algorithm that attempts to construct the adjacency matrix of a strongly regular graph (given a suitable order-free design) is presented. Two appendices related to the algorithm are included. The first lists all parameter sets (n,a,c,d) with n ≤ 50 and a ≠ d that satisfy the standard feasibility conditions for strongly regular graphs. Additional information is included for each set. The second appendix contains adjacency matrices (with the partitioning by cocliques and order-free designs exhibited) for most of the parameter sets in the first appendix. The theoretical development is presented in the context of association schemes. Partitioning by order-free designs extends naturally to any association scheme when cocliques are generalized to {Ø,i} -cliques. This extended partitioning is applied to generalized hexagons.

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