The terwilliger algebra of a distance-regular graph


Junie T. Go

Date of Publication


Document Type


Degree Name

Doctor of Philosophy in Mathematics

Subject Categories



College of Science


Mathematics and Statistics


This dissertation deals with the Terwilliger algebra of a distance-regular graph.The study has two main parts. The first part studies the Terwilliger algebra of the D-cube QD, also known as hypercube. Let X denote the vertex set of QD. Fix x e X, and let T=T(x) denote its associated Terwilliger algebra. T is shown as the subalgebra of Matx (C) generated by the adjacency matrix A and a diagonal matrix A*=A*(x), where A* has yy entry D-2a(x,y) for all y e X. A, A* satisfyA2A*-2AA*+A*A2 = 4A*,A*2-2A*AA*+AA*2 = 4AUsing the above equations, the irreducible T-modules is found. For each irreducible T-module W, two orthogonal bases are displayed, the standard basis and the dual standard basis. Action of A and A* are described on these basis. The transition matrix is given from the standard basis to the dual standard basis. The multiplicity with which each irreducible T-module W appears in is computed. An elementary proof that QD has the Q-polynomial property is given. T, a homomorphic image of the universal enveloping algebra of the Lie algebra sl2 (C) is shown. The center of T is described.

The second part of this dissertation studies the Terwilliger algebra of a tight distance-regular graph. Let r = (X,R) denote a distance-regular graph with diameter D greater than or equal to 3. Fix x e C, and let T - T(x) denote its associated Terwilliger algebra. We associate two integer parameters: the endpoint and the diameter, to each irreducible T-module. It turns out that the dimension of such a module is at least one more than its diameter. Whenever equality is attained, the module is said to be thin. To each irreducible T-module of endpoint 1 and diameter D-2, another real parameter, the type is associated. The assumption now is that r is tight. The r is shown to have at least one irreducible T-module of type 1, at least one irreducible T-module of type D, and up to isomorphism, no other irreducible T-modules of endpoint 1. Each type is shown to be thin and has diameter D-2. The multiplicity with which each module appears inCx is computed.

Abstract Format






Accession Number


Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

Physical Description

165 leaves ; Computer print-out


Graph theory; Algebra; Polynomials

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