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# Symmetric bicirculants

Gorazd Vasiljević (2014) Symmetric bicirculants. Diploma thesis.

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## Abstract

This BSc thesis deals with certain topics from graph theory. When we talk about studying graphs, we usually mean studying their structure and their structural properties. By doing that, we are often interested in automorphisms of a graph (symmetries), which are permutations of its vertex set, preserving adjacency. There exist graphs, which are symmetric enough, so that automorhism group acts transitively on their vertex set. This means that for any pair of vertices of the graph, there is an automorphism, mapping one vertex to the other. Such graphs are called vertex-transitive. Similarly, a graph is edge-transitive, if its automorphism group acts transitively on its edge set and is arc-transitive, if its automorphism group acts transitively on its arc set. A bicirculant is a graph, which admits an automorhpism with two orbits of the same length. When studying bicirculants, one of the goals is to classify or at least identify infinite families of vertex-, edge- or arc-transitive bicirculants, given additional restrictions, such as the degree of vertices. In this thesis, we will examine the results of Frucht, Graver and Watkins, who studied the so-called Generalised Petersen graphs (cubic bicirculants) and classified the vertex- and edge-transitive ones. The next natural step is to study the natural generalizations of Generalised Petersen graphs, the so-called Rose-window graphs. The first to study them was Wilson, who also identified four families of edge-transitive Rose-window graphs. The main focus of this thesis is on Wilson's work. We present four families of Rose-Window graphs from Wilson's paper and show, that their members are in fact edge-transitive bicirculants.

Item Type: Thesis (Diploma thesis)
Keywords: graph theory, bicirculant, automorphism, Rose-Window graph, edge-transitive graph
Number of Pages: 39
Language of Content: Slovenian
Mentor / Comentors:
Mentor / ComentorsIDFunction
doc. dr. Primož ŠparlMentor
Link to COBISS: http://www.cobiss.si/scripts/cobiss?command=search&base=50126&select=(ID=10289225)
Institution: University of Ljubljana
Department: Faculty of Education
Item ID: 2597
Date Deposited: 07 Nov 2014 14:11