Title

On central elements in the Terwilliger algebra of hamming graph

Date of Publication

2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Adviser

Arlene A. Pascasio

Defense Panel Chair

Ederlina G. Nocon

Defense Panel Member

Rafael Reno S. Cantuba
Reginaldo M. Marcelo
Edmundo D. Perez, Jr.
Melvin A. Vidar

Abstract/Summary

Let n and D be positive integers with n 3, and let H(D n) denote the Hamming graph. Recall the graph H(D n) is distance-regular of diameter D. Let X denote the vertex set of H(D n), and let MatX(C) denote the C-algebra of matrices with rows and columns indexed by X. Let A 2 MatX(C) denote the adjacency matrix and let @ denote the path-length distance in H(D n).

Fix x 2 X. For all i (0 i D), let E i = E i (x) 2 MatX(C) denote the diagonal matrix with yy-entry equal to 1 if @(x y) = i and 0 otherwise, for y 2 X. Let T = T(x) denote the subalgebra of MatX(C) generated by A and E 0 E 1 : : : E D of H(D n). We call T the Terwilliger algebra of H(D n) with respect to x. It is known that A and A generate T, where A = PD i=0 i E i and i = (n {u100000} 1)D {u100000} ni (0 i D). By the center of T, denoted Z(T), we mean the subalgebra of T consisting of elements that commute with all elements of T.

This study focuses on describing all elements C 2 Z(T) of H(D n) satisfying the property that for all y z 2 X with @(y z) 2, the yz-entry of C is 0. We show that C = XD i=0 iE i AE i + XD i=0 iE i for some i i 2 C for 0 i D. We determine the scalars i and i. Finally, we prove the conjecture of Terwilliger that the space of all central elements satisfying the given property has basis f Ig where = n(n {u100000} 2)A + n2A {u100000} A2A + 2AA A {u100000} A A2 2n(n {u100000} 1) and I is the identity matrix in MatX(C).

Abstract Format

html

Language

English

Format

Electronic

Accession Number

CDTG007629

Shelf Location

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

Physical Description

1 computer disc; 4 3/4 in.

Keywords

Graph theory

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