On the Action of ?0(N) on ??

In this paper we examine ?0(N)-orbits on ?? and the suborbital graphs for ?0(N). Each such suborbital graph is a disjoint union of subgraphs whose vertices form a block of imprimitivity for ?0(N). Moreover, these subgraphs are shown to be vertex ?0(N)-transitive and edge ?0(N)-transitive. Finally, necessary and sufficient conditions for being self-paired edge are provided. © 2010 Università del Salento

On conjugacy of elliptic elements and circuits in suborbital graphs of congruence subgroups

In this paper, we, for the first time, introduce the notion of conjugacy in suborbital graphs. Then we show that the conjugate elliptic elements of the modular group F and of congruence subgroups F 0(p) give rise to conjugate circuits, corresponding to the related elliptic elements, in the Farey graph F and in the suborbital graph F U,p of the action of F 0(P), respectively

On suborbital graphs and related continued fractions

In this paper, we study suborbital graphs for congruence subgroup ?0(n) of the modular group ? to have hyperbolic paths of minimal lengths. It turns out that these graphs give rise to a special continued fraction which is a special case of very famous fraction coming out from Pringsheim's theorem. © 2011 Elsevier Inc. All rights reserved

Elliptic elements and circuits in suborbital graphs

We consider the action of a permutation group on a set in the spirit of the theory of permutation groups, and graph arising from this action in hyperbolic geometric terms. In this paper, we examine some relations between elliptic elements and circuits in graph for the normalizer of ?0(N) in PSL(2,?)

Suborbital graphs for the group ?2

In this paper, we investigate suborbital graphs formed by the action of ?2 which is the group generated by the second powers of the elements of the modular group ? on Q. Firstly, conditions for being an edge, self-paired and paired graphs are provided, then we give necessary and sufficient conditions for the suborbital graphs to contain a circuit and to be a forest. Finally, we examine the connectivity of the subgraph Fu, N and show that it is connected if and only if N ? 2. © 2015, Hacettepe Journal of Mathematics and Statistics. All rights reserved