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Terwilliger, Paul M. 2014-12-16T06:18:03Z 2014-12-16T06:18:03Z 10000-01-01 1982 1982 A distance-transitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PAR-DIFF)(u,v) = (PAR-DIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distance-transitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distance-regular. We deal first with distance-regular graphs with girth 3 and 4. We show any such graph which contains a cycle (v(,1),v(,2),v(,3),v(,4),v(,1)) where (PAR-DIFF)(v(,1),v(,3)) = (PAR-DIFF)(v(,2),v(,4)) = 2 is finite with diameter d bounded by its valency k. Next we obtain new "feasibility conditions" that the intersection numbers of arbitrary distance-regular graphs with girth 3 or 4 must satisfy. We use these conditions to generate the intersection array of certain distance-regular graphs from their valency and three other parameters. We then study distance-regular graphs with arbitrary girth and extend the ideas used in the girth 3 or 4 case to obtain a bound on the diameter of a class of distance-regular graphs, including all those with even girth. We then introduce a generalization of a distance-regular graph called a (s,c,a,k)-graph, which possesses enough of the local structure of a distance-regular graph to enable us to find bounds on their diameters in some cases. Finally we obtain lower bounds on the eigenvalue multiplicities for distance-regular graphs in terms of their valence and girth, yielding additional feasibility conditions for their intersection arrays. Made available in DSpace on 2014-12-16T06:18:03Z (GMT). No. of bitstreams: 1 8218574.pdf: 2612218 bytes, checksum: a2c414f514218467b97cd3b109618e49 (MD5) Previous issue date: 1982 Embargo set by: Seth Robbins for item 71370 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 115 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982. http://hdl.handle.net/2142/71204 (UMI)AAI8218574 Mathematics Distance-Regular Graphs and Generalizations text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.