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Title: Geodetic graphs and related concepts
Author: Pryce, D. G.
Awarding Body: University College of Swansea
Current Institution: Swansea University
Date of Award: 1980
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The question of the characterisation of graphs in which each pair of vertices is joined by a unique geodesic was first raised by Oystein Ore. Such graphs were termed geodetic graphs and the first published account of a partial solution to this problem was given by Stemple and Watkins who characterised planar geodetic graphs. Various writers (see the references in the text) have dealt with geodetic graphs of diameter two. After a note on terminology and an introduction in Chapter I we proceed in Chapter II to give an account of discussions by Plesnik and Stemple of the geodetic homeomorphs of a complete graph on any number of vertices. In this account some of the proofs are shortened and the calculation of the diameters of such graphs together with some enumeration questions are discussed. In Chapter III we give the construction of a new class of graphs based on Plesnik's geodetic homeomorphs of a complete graph and show that these new graphs are geodetic. Examples are given to illustrate the variety of types of such graph and to show the calculation of their diameters. Chapter IV shows how this family of geodetic graphs may be considerably extended and gives examples. We prove that this extended family of graphs includes a class of geodetic graphs constructed by Plesnik. In conclusion Chapter V discusses graphs for which the number of geodesics between two vertices depends only on the distance apart of the vertices. These graphs include some symmetric regular graphs given by Biggs but we show that there exist classes of such graphs which are not regular.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available