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Abstract
This thesis provides a study of selfavoiding walks on quasitransitive graphs. The connective constant µ(G) of a graph G is the asymptotic growth rate of the number of selfavoiding walks on G starting at a given vertex. For a given graph height function mapping vertices of G to integers in a way adapted to the graph structure, a bridge is a selfavoiding walk such that the height of its vertices is bounded below by the height of the initial vertex and above by the height of the terminal vertex. If the roles of the initial and terminal vertices are reversed, we talk about reversed bridges. We show that for any graph height function, the maximum of the asymptotic growth rates of the number of bridges and the number of reversed bridges must be equal to the connective constant µ(G).
The main focus of this thesis is to apply the theory of formal languages to the study of selfavoiding walks. To this end, let G be a deterministically edgelabelled graph, that is, every (directed) edge carries a label such that any two edges starting at the same vertex have diﬀerent labels. Then the set of all words which can be read along the edges of selfavoiding walks starting at o forms a language denoted by LSAW,o(G). We show that the properties of this language strongly depend on the endstructure of the graph G. It is regular if and only if all ends have size 1 and it is contextfree if and only if all ends have size at most 2.
Making use of the class of multiple contextfree languages, this characterisation can be extended even further. We show that LSAW,o(G) is a kmultiple contextfree language if and only if the size of all ends of G is at most 2k. Applied to Cayley graphs of ﬁnitely generated groups this says that LSAW,o(G) is multiple contextfree if and only if the group is virtually free. In this setting, using our method we also show that the ordinary generating function of selfavoiding walks is algebraic and in particular, the connective constant is an algebraic number.
The main focus of this thesis is to apply the theory of formal languages to the study of selfavoiding walks. To this end, let G be a deterministically edgelabelled graph, that is, every (directed) edge carries a label such that any two edges starting at the same vertex have diﬀerent labels. Then the set of all words which can be read along the edges of selfavoiding walks starting at o forms a language denoted by LSAW,o(G). We show that the properties of this language strongly depend on the endstructure of the graph G. It is regular if and only if all ends have size 1 and it is contextfree if and only if all ends have size at most 2.
Making use of the class of multiple contextfree languages, this characterisation can be extended even further. We show that LSAW,o(G) is a kmultiple contextfree language if and only if the size of all ends of G is at most 2k. Applied to Cayley graphs of ﬁnitely generated groups this says that LSAW,o(G) is multiple contextfree if and only if the group is virtually free. In this setting, using our method we also show that the ordinary generating function of selfavoiding walks is algebraic and in particular, the connective constant is an algebraic number.
Original language  English 

Qualification  Doctor of Technology 
Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  8 Sept 2021 
Publication status  Published  Jul 2021 
Keywords
 Selfavoiding walks
 Multiple context free language
ASJC Scopus subject areas
 Discrete Mathematics and Combinatorics
 Theoretical Computer Science
Fields of Expertise
 Information, Communication & Computing
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 1 Active

Doctoral Program: Discrete Mathematics
Ebner, O., Lehner, F., Greinecker, F., Burkard, R., Wallner, J., Elsholtz, C., Woess, W., Raseta, M., Bazarova, A., Krenn, D., Lehner, F., Kang, M., Tichy, R., SavaHuss, E., Klinz, B., Heuberger, C., Grabner, P., Barroero, F., Cuno, J., Kreso, D., Berkes, I. & Kerber, M.
1/05/10 → 30/06/24
Project: Research project