Flag graphs and symmetry type graphsRío Francos, María del (Avtor) Pisanski, Tomaž (Mentor) Hubard, Isabel (Komentor) flag graphsymmetry type graphk-orbit mapmaniplexmedialchamferingtruncationleapfrogA map, as a 2-cell embedding of a graph on a closed surface, is called a ▫$k$▫-orbit map if the group of automorphisms (or symmetries) of the map partitions its set of flags into ▫$k$▫ orbits. In 2012, Steve Wilson introduced the concept of maniplex, aming to unify the notion of maps and abstract polytopes. In particular, maniplexes generalise maps on surfaces to higher ranks. The combinatorial structure of a maniplex of rank ▫$(n-1)$▫ (or an ▫$(n-1)$▫-maniplex) is completely determined by an edge-coloured ▫$n$▫-valent graph with chromatic index ▫$n$▫, with ▫$n \ge 1$▫, often called the flag graph of the maniplex. Maps will be regarded as maniplexes of rank 2 (or 2-maniplexes), and defined as Lins and Vince studied the combinatorial maps since 1982-1983. Thus, similarly to maps, a ▫$k$▫-orbit maniplex is one that has ▫$k$▫ orbits of flags under the action of its automorphism group. In the first part of this thesis we introduce the notion of symmetry type graphs of maniplexes and make use of them to study ▫$k$▫-orbit maniplexes, as well as fully-transitive 3-maniplexes. We classify all possible symmetry types of ▫$k$▫-orbit 2-maniplexes for ▫$k \le 5$▫, as well as all self-dual, properly and improperly, ▫$k$▫-orbit maps with ▫$k \le 7$▫. Moreover, we show that there are no fully-transitive ▫$k$▫-orbit 3-maniplexes with ▫$k > 1$▫ an odd number, we classify 3-orbit maniplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Furthermore, we give generators of the automorphism group of a maniplex, given its symmetry typegraph. The second part of this work is motivated by the classification for ▫$k$▫-orbit, up to ▫$k \le 4$▫, that Orbanić, Pellicer and Weiss gave. Thus, motivated by their results, we use symmetry type graphs to extend such study and classify all the types of ▫$k$▫-orbit maps with the same operations on maps, up to ▫$k \le 6$▫. Furthermore, we studied other operations on maps, such as the chamfering and leapfrog operations. In particular, we determine all possible symmetry types of maps that result from other maps after applying the chamfering operation and give the number of possible flag-orbits that has the chamfering map of a ▫$k$▫-orbit map.[M. del Río Francos]20142017-09-22 02:53:40Doktorsko delo/naloga95851sl