The A-like matrices for hypercube and cycle

Date of Publication

2013

Document Type

Master's Thesis

Degree Name

Master of Science in Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics

Abstract/Summary

The paper is composed of two parts. The first part is an exposition of the article The A- like matrices for a hypercube by Stefco Miklavi and Paul Terwilliger which appeared in the Electronic Journal of Linear Algebra ISSN 1081-3810, Volume 22, pp. 796-809 last Augut 2011. In the second part, as inspired by the above mentioned paper, we give the A-like matrices for a cycle. Let {u100000}(X R) denote a distance-regular graph and let A 2 MatX(R) denote the adjacency matrix of {u100000}. We define a matrix B 2 MatX(R) to be A-like whenever both (i) BA = AB and (ii) for all x y 2 X that are not equal or adjacent, the (x y)-entry of B is zero. Let L denote the subspace of MatX(R) consisting of the A-like elements. Then L is decomposed as direct sum of its symmetric part and antisymmetric part denoted as Lsym and Lasym, respectively. Let D denote a positive integer and let QD denote the D-dimensional hypercube. For {u100000} = QD, Miklavic and Terwilliger in [6] found a basis for Lsym and Lasym and showed that the dimensions are D + 1 and {u100000}D 2 respectively. Let k 3 denote an integer and let Ck denote the cycle with k vertices. Observe that the cycle C4 is isomorphic to Q2. For {u100000} = C3, we found that Lsym has dimension 4 while Lasym has dimension 1. For k 5, we found a basis for Lsym and Lasym and showed that their dimensions are 2 and 1, respectively.

Abstract Format

html

Language

English

Format

Electronic

Accession Number

CDTG005327

Shelf Location

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

Physical Description

1 computer optical disc ; 4 3/4 in.

Keywords

Hypercube

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