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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 Urbana-Champaign
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 Urbana-Champaign, 1987.
URI:http://hdl.handle.net/2142/71261
Other Identifier(s):(UMI)AAI8803208
Date Available in IDEALS:2014-12-16
Date Deposited:1987


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