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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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