Files in this item
Files  Description  Format 

application/pdf 8803208.pdf (3MB)  (no description provided) 
Description
Title:  Automorphism Groups of the Augmented Distance Graphs of Trees 
Author(s):  Sportsman, Joseph Scott 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\Gamma\sb{\rm i}$, and their associated groups. In this thesis we introduce the graphs, $\Gamma\sp{\rm (r)}$ defined by $\Gamma\sp{\rm (r)}$ = $\Gamma\sb1$ + $\Gamma\sb2$ + $\cdots$ + $\Gamma\sb{\rm r}$ and their automorphism groups G$\sp{\rm (r)}$. We show that for a tree $\Gamma$, the groups G$\sp{\rm (r)}$ form a tower which is not the case for arbitrary graphs. From this, we give a description of the structure of G$\sp{\rm (r)}$ for trees and completely characterize the trees of a fixed diameter which have minimal group tower length. Also we introduce a new parameter, $\chi$ for trees defined as follows: Let x and y be vertices of $\Gamma$. Partition the remaining vertices into three sets; W(x) = $\{$w$\epsilon$V($\Gamma$): $\partial$(w,x)$$ 0$\}$. It turns out that $\chi$ has nice properties. One theorem we prove is the following: If $\Gamma$ is a tree of diameter greater than 3, and m = min$\{\chi$ + 1, (d/2) $\}$, then G$\sp{\rm (m+1)}$ $\not=$ G, but G$\sp{\rm (r)}$ = G for all r $\leq$ m. 
Issue Date:  1987 
Type:  Text 
Description:  92 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/71261 
Other Identifier(s):  (UMI)AAI8803208 
Date Available in IDEALS:  20141216 
Date Deposited:  1987 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois