In this talk we present the main topics, research questions and the expected results of the proposed PhD thesis.

The central theme of the PhD thesis are graphs admitting a considerable degree of symmetry. More precisely, we focus on graphs admitting a vertex- and edge-transitive group of automorphisms. 

A graph Γ is said to be G-vertex-transitive, G-edge-transitive and G-arc-transitive whenever the subgroup G ≤ Aut(Γ) acts transitively on V(Γ), E(Γ) and A(Γ), respectively. We say that Γ is G-half-arc-transitive (abbreviated by G-HAT) if it is G-vertex- and G-edge-transitive but not G-arc-transitive. In the case of G=Aut(Γ), we omit the prefix G and simply write vertex-transitive, edge-transitive, arc-transitive and half-arc-transitive (abbreviated by HAT).

Let Γ be a G-vertex- and G-edge-transitive graph for some G ≤ Aut(Γ). Then two essentially different possibilities can occur:
(i) Γ is G-arc-transitive.
(ii) Γ is G-half-arc-transitive.

In the first main topic of the PhD thesis we will focus on the situations from the above possibility (i). In particular, we will be interested in the application of the properties of arc-transitive graphs as a tool in the investigation of symmetries of maps. 

In the second and third part of the PhD thesis, we will focus on the situations from the above possibility (ii). We plan to introduce a new parameter of tetravalent G-HAT graphs, giving a better understanding of the structural properties of such graphs. We will study the properties of the graphs with respect to this parameter and use it to relate two important approaches for a possible classification of all tetravalent G-HAT graphs. We will also improve the results on the question of whether the attachment number divides the radius for all tetravalent HAT graphs. 

Finally, we will focus on HAT graphs with valencies greater than four. We will generalize the Bouwer graphs to obtain a much larger family of vertex- and edge-transitive graphs, most of whose members are in fact tightly attached HAT graphs.