Homotopy in graphs

Date of Publication

1999

Document Type

Master's Thesis

Degree Name

Master of Science in Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics

Thesis Adviser

Arlene A. Pascasio

Defense Panel Chair

Yolando B. Beronque

Defense Panel Member

Blessilda P. Raposa
Junie T. Go

Abstract/Summary

This thesis is an exposition of Sections 1 to 7 of the article entitled Homotopy in Q-Polynomial Distance-Regular Graphs by Heather A. Lewis submitted to Discrete 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 x in G, and consider the set u/(x) of all closed paths in G with base vertex x. We define a relation on this set, called homotophy, and prove that it is an equivalence relation. We denote the equivalence classes by minimum degree (x) and show that path concatenation induces a group structure on minimum degree (x). We define essential length of an element in minimum degree (x), and using this concept, we define a collection of subgroups minimum degree (x,i).Now, suppose x, y are vertices in G, and suppose there exists a path from x to y. We show the groups minimum degree (x) and minimum degree (y) are isomorphic, and that the isomorphism preserves essential length.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 x, minimum degree (x,2d+1)=minimum degree (x).

Abstract Format

html

Language

English

Format

Print

Accession Number

TG02935

Shelf Location

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

Physical Description

79 leaves

Keywords

Graph theory; Homotopy theory; Topology

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