Suborbital graphs of Hecke groups ?^3 and g_5

In this thesis, the main object is to examine the relationships between the lengths of circuits of suborbital graphs arising from the action of the subgroup ?^3 of modular group ? and Hecke group G_5 on Q?{?} and Q[?]?{?} respectively and orders of generating elliptic elements of these groups. And furthermore connectedness of suborbital graphs for the group ?^3 is presented.In Chapter 1, the structure of Non-Euclidean crystalllographic groups is discussed and some properties of PSL(2,R), the modular group ?, congruence subgroups of ? and also the preliminary definitions we require for imprimitive action, graph theory, Hecke groups and algebraic number fields are given.In Chapter 2, two different problems are discussed. Firstly, suborbital graphs arising from modular subgroup ?^3 are examined. And then edge and circuit conditions on these graphs are determined and the connectedness of graphs are examined under these conditions. Secondly, suborbital graphs arising from Hecke group G_5 are examined and edge conditions of these graphs are obtained under some certain conditions.