{"id":{"repo_id":"durham","oai_identifier":"oai:etheses.durham.ac.uk:272"},"canonical_url":"https://search.dev.ndltd.org/etd/durham/oai:etheses.durham.ac.uk:272","repository":{"repo_id":"durham","name":"Durham University","base_url":"http://etheses.dur.ac.uk/cgi/oai2"},"display":{"title":"Counting and Averaging Problems in Graph Theory","abstract":"Paul Gunther (1966), proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. Heinz Huber (1956) considered the following problem on the hyperbolic plane H: Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. In this thesis, we use a well-known analogy between the hyperbolic plane and the regular tree to solve the above problems, and some related ones, on a tree. We deal mainly with regular trees, however some results incorporate more general graphs.","abstract_html":"Paul Gunther (1966), proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. Heinz Huber (1956) considered the following problem on the hyperbolic plane H: Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. In this thesis, we use a well-known analogy between the hyperbolic plane and the regular tree to solve the above problems, and some related ones, on a tree. We deal mainly with regular trees, however some results incorporate more general graphs.","abstract_has_math":false,"creators":["Douma, Femke"],"institution":"Durham University","degree_name":"PhD","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-24T02:11:04Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Douma, Femke"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010"]},{"key":"dc:date.issued","label":"Date","values":["2010"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematical Sciences, Department of"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Durham University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.durham.ac.uk/id/eprint/272/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.durham.ac.uk/id/eprint/272/1/thesisFemkeDouma.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Paul Gunther (1966), proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. Heinz Huber (1956) considered the following problem on the hyperbolic plane H: Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. In this thesis, we use a well-known analogy between the hyperbolic plane and the regular tree to solve the above problems, and some related ones, on a tree. We deal mainly with regular trees, however some results incorporate more general graphs."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["Counting and Averaging Problems in Graph Theory"]}]}],"canonical_facts":{"dc:creator":["Douma, Femke"],"dc:date":["2010"],"dc:date.issued":["2010"],"dc:description.abstract":["Paul Gunther (1966), proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. Heinz Huber (1956) considered the following problem on the hyperbolic plane H: Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. In this thesis, we use a well-known analogy between the hyperbolic plane and the regular tree to solve the above problems, and some related ones, on a tree. We deal mainly with regular trees, however some results incorporate more general graphs."],"dc:format":["text"],"dc:identifier.uri":["https://etheses.durham.ac.uk/id/eprint/272/1/thesisFemkeDouma.pdf"],"dc:publisher.department":["Mathematical Sciences, Department of"],"dc:publisher.institution":["Durham University"],"dc:relation.isreferencedby":["https://etheses.durham.ac.uk/id/eprint/272/"],"dc:title":["Counting and Averaging Problems in Graph Theory"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-24T02:11:04Z"}