## Dissertations

#### Title

On central elements in the Terwilliger algebra of hamming graph

2018

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

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).

html

English

Electronic

CDTG007629

#### Shelf Location

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

#### Physical Description

1 computer disc; 4 3/4 in.

Graph theory

COinS