A point-weighted structure is an incidence structure with each point assigned an element of some set W C Z+ as a 'weight'. A point-weighted structure with no repeated blocks and the property that the sum of the weights of the points incident with anyone block is a constant k is called a point-weighted design. A t - (v, k, Aj W) point-weighted design is such a structure with the sum of the weights of all the points equal to v and the property that every set of t distinct points is incident with exactly A blocks. This thesis introduces and examines this generalisation of block designs. The first chapter introduces incidence structures and designs. Chapter 2 introduces and defines point-weighted designs. Three constructions of families of t - (v, k, Aj W) point-weighted designs are given. Associated with any point-weighted design is the incidence structure on which it is based - the 'underlying' incidence structure (u.i.s.). It is shown in Chapter 3 that any automorphism of the u.i.s. of a t - (v, k, Aj W) point-weighted design with more than one block and t > 1 preserves weights in the point-weighted design. The u.i.s. of such a point-weighted design is shown to be a block design if and only if every point is assigned the same weight. A necessary and sufficient condition is obtained for the assignment of weights in any point-weighted design to be essentially uniquely determined by the u.i.s. Chapter 4 considers t-{v, k, Aj W) point-weighted designs in which all of the points apart from a 'special' point have the same weight. It is shown that when v > k the weight of the special point is an integer multiple of the weight assigned to all the other points. A class of these point-weighted designs is demonstrated to be equivalent to a class of group-divisible designs with specific parameters. The final chapter uses the procedure of point-complementing incidence structures to construct point-weighted designs. Trivial point-weighted designs are defined and a necessary and sufficient condition for the existence of a member of a certain class of these is obtained. A correspondence between this class of point-weighted designs and certain trivial block designs is given using point-complementing.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:243571 |
| Date | January 1995 |
| Creators | Horne, Richard Brian Denison |
| Publisher | Royal Holloway, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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