Felix L.;p"ao

i4 'l{ 6s7

+g

fll'jc6

@ ^{De I^a Salle University}

Horuoropy IN Gnapns

A Thesis Presented to

The Faculty of the Mathematics Department College of

Science

De La Salle University

In Partial Fulfillment

of the Requirements for the

Degree

Master of

Science

in Mathematics

by

Felix L.;p"ao

August

1999

@ ^{De I^a salle university}

Abstract

This thesis is an exposition of Sections

l

to 7 of the article entitled "Homotopy in Q-Polynomial Distance-Regular Graphs" by Heather A. Lewis submitted to Dis- crete Mathematics. The aforementioned article constitutes the first two chapters

of Lewis' dissertation "Homotopy and Distance-Regular Graphs", University of Wisconsin Madison, U.S.A, ^1997.

Let G

denote an undirected graph

without

loops

or

multiple edges.

Fix

a vertex

r

in G, and consider the set

/(r)

^{of all closed paths in G with base vertex}

r.

^{We define} a relation on

this

set, called homotopy, and ^prove

that it

is an equivalence

relation.

We denote th-e equivalence classes by

zr(r)

and show that path concatenation induces a group structure on

z'(r).

We define essential length of an element in zr'(r), and using this concept, we define a collection of subgroups

r(r,i).

i

Now, suppose ,r, ^gr are vertices in G, and suppose there exists a path from c to

g.

We show the ^groups zr(u) and

r(y)

arc isomorphic, and

that

the isomorphism preserves essential length.

@ ^{De Ia Salle university}

We define a geodesic path and prove a sufficient condition for a path

to

be geodesic. Finally, we assume G is finite and connected with diameter d. We find an upper bound for the length of a geodesic closed path. We prove that for any

fix vertex

r, r(r,2d + l) : _r(r).