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Vertex-transitive graphs and maps and their automorphism groups
Abstract
The first part of this dissertation deals with highly symmetrical combinatorial structures - vertex transitive graphs. One of the best known examples of infinite families of vertex-transitive graphs are the Cayley graphs. Besides these, and some additional sporadic examples, not many examples have been known. Quite recently, a great deal of activity has brought us a considerable number of new interesting examples of vertex-transitive graphs that are not Cayley but the work is far from being done. Constructions introduced in the dissertation are based on a generalization of the Cayley graph construction. This allows one to construct all vertex-transitive graphs. Using an additional graph-theoretic condition, necessarily satisfied by all Cayley graphs, we are able to prove the graphs we obtain are not Cayley. The second part of my dissertation is devoted to combinatorial structures living on the border between topological graph theory and algebraic combinatorics--Cayley maps. Being 2-cell embeddings of Cayley graphs by their very nature, Cayley maps are topological objects. My study of them is based on a combinatorial description that allows me to handle the basic question of automorphism groups of Cayley maps in a manner similar to methods used in the first part. We provide a full description of these automorphism groups, based on the concept of a rotary mapping. This description allows one to solve several related questions, in particular, a question posed by Biggs and White in 1979 about the characterization of Cayley maps with regular automorphism groups that has not been resolved for over twenty years. The dissertation also includes a new construction of regular Cayley maps and a characterization of finite groups admitting regular Cayley maps.
Subject Area
Mathematics
Recommended Citation
Jajcay, Robert, "Vertex-transitive graphs and maps and their automorphism groups" (1995). ETD collection for University of Nebraska-Lincoln. AAI9538634.
https://digitalcommons.unl.edu/dissertations/AAI9538634