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Title:  DistanceRegular Graphs and Generalizations 
Author(s):  Terwilliger, Paul M. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  A distancetransitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PARDIFF)(u,v) = (PARDIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distancetransitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distanceregular. We deal first with distanceregular 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 (PARDIFF)(v(,1),v(,3)) = (PARDIFF)(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 distanceregular graphs with girth 3 or 4 must satisfy. We use these conditions to generate the intersection array of certain distanceregular graphs from their valency and three other parameters. We then study distanceregular 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 distanceregular graphs, including all those with even girth. We then introduce a generalization of a distanceregular graph called a (s,c,a,k)graph, which possesses enough of the local structure of a distanceregular graph to enable us to find bounds on their diameters in some cases. Finally we obtain lower bounds on the eigenvalue multiplicities for distanceregular graphs in terms of their valence and girth, yielding additional feasibility conditions for their intersection arrays. 
Issue Date:  1982 
Type:  Text 
Description:  115 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1982. 
URI:  http://hdl.handle.net/2142/71204 
Other Identifier(s):  (UMI)AAI8218574 
Date Available in IDEALS:  20141216 
Date Deposited:  1982 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois