Title:

Some graphical problems related to score sequences, partial orders and finite topologies

This thesis presents a study of some topics and problems of graph theory that are related to partial orders. It is an attempt to develop and utilise the links between score sequences of oriented graphs, different types of partial orders, and combinatorial aspects of finite topologies. In the first chapter Landau's concept of the score structure of a tournament is extended to that of the score sequence of an oriented graph, and the condition is given for an arbitrary integer sequence to be the score sequence of some oriented graph. The proof involves the construction of a specific oriented graph D(S) with the given score sequence S. Landau's condition for the score structure of a tournament is deduced as a simple consequence by constructing a tournament T(L) with given score structure L. It is shown that D(S) is a partial order for every S, and an enumeration of score sequences /labelled score sequences provides a lower bound for the number of partial orders /T* topologies. The partial order D(S) has the minimum number of arcs among the oriented graphs with score sequence S, and T(L) has the maximum number of upsets among the tournaments with score structure L. We also establish that given two oriented graphs with the same score sequence, one can be transformed to the other by successively transforming appropriate intransitive triples to transitive triples or vice versa. The proof of an analogous earlier result of Ryser for tournaments reduces to a particular case of the proof of this result. The second chapter begins with an introduction to some of the ideas of paired comparison experiments. If indifference is not permitted then the preference pattern induced by such an experiment is a tournament, and a consistent preference pattern is a linear order. If indifference is permitted then the Induced preference pattern is an oriented graph, and one interpretation of consistency is that indifference as well as preference should be transitive, in which case consistent preference patterns are weak orders. We show that those oriented graphs which are specified up to isomorphism by their score sequences are weak orders, and establish some further results relating weak orders and score sequences. In a similar vein, the condition is given for a tournament to be specified up to isomorphism by its score structure. The third chapter is about semiorders, which, like weak orders, are special types of partial orders. Luce and others have suggested that preferences in paired comparison experiments become recognisable only when of sufficient magnitude. Under this assumption, consistent preference patterns are characterised as semiorders. The proof relies upon reformulating the definition of a semiorder in terms of its score sequence. There is an interesting application of this result to significance testing in statistics. The number of semiorders with n vertices is shown to be the nth Catalan number and labelled semiorders are counted. Harary's reconstruction conjecture is proved for semiorders. Then we consider extensions of the Kendall and Babington Smith definition of the coefficient of consistency of a paired comparison experiment to those experiments in which indifference is permitted. In the final chapter, the links between finite topologies and transitive digraphs are studied. It is well Known that there is a natural one toone correspondence between the finite topologies and transitive digraphs with the same point (vertex) set. This correspondence is exploited to derive the graphical analogues of topological properties such as connectivity, maximal connectivity, fineness, the TQ separation axiom and closure. Certain binary operations between topologies which produce other topologies, such as union. Intersection and the cartesian product, are also shown to have simple analogues in graph theory. The number of npoint maximal connected topologies is counted and related to the counting series for rooted trees. Then the following problem is investigated: given n, for which values of r is there an npoint topology with r open sets? The cardinalities of topologies are studied by considering their associated digraphs, and it transpires that it is necessary to consider only those transitive digraphs which are partial orders. Various results are proved, including existence and nonexistence criteria, enabling a complete solution to be given for n ^ 9. In conclusion, three conjectures about cardinalities are put forward.
