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.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/187829 |
| Date | January 1984 |
| Creators | GOSSETT, ERIC JAMES. |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Dissertation-Reproduction (electronic) |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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