Triangles in the suborbital graphs of the normalizer of $\Gamma_0(N)$

<p>In this paper, we investigate a suborbital graph for the normalizer of Γ_{0(<em>N</em>)} ∈ PSL(2;<em>R</em>), where <em>N</em> will be of the form 2⁴<em>p</em>² such that <em>p</em> &gt; 3 is a prime number. Then we give edge and circuit conditions on graphs arising from the non-transitive action of the normalizer.</p>

Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups

The action of affine groups on Galois field has been studied. For instance, studied the action of on Galois field for a power of prime. In this paper, the rank and subdegree of the direct product of affine groups over Galois field acting on the cartesian product of Galois field is determined. The application of the definition of the product action is used to achieve this. The ranks and subdegrees are used in determination of suborbital graph, the non-trivial suborbital graphs that correspond to this action have been constructed using Sims procedure and were found to have a girth of 0, 3, 4 and 6.

Suborbital graphs for a non-transitive action of the normalizer

In this paper, we investigate a suborbital graph for the normalizer of ?0(n) in PSL(2,R), where n will be of the form 32p2, p is a prime and p > 3. Then we give edge and circuit conditions on graphs arising from the non-transitive action of the normalizer.

On Suborbital Graphs for the Normalizer of $\Gamma_{0}(N)$

In this study, we deal with the conjecture given in [R. Keskin, Suborbital graph for the normalizer of $\Gamma _{0}(m)$, European Journal of Combinatorics 27 (2006) 193-206.], that when the normalizer of $\Gamma _{0}(N)$ acts transitively on ${\Bbb Q\cup\{\infty \}}$, any circuit in the suborbital graph $G(\infty,u/n)$ for the normalizer of $\Gamma _{0}(N),$ is of the form $$ v\rightarrow T(v)\rightarrow T^{2}(v)\rightarrow {\ \cdot \cdot \cdot } \rightarrow T^{k-1}(v)\rightarrow v, $$ where $n>1$, $v\in {\Bbb Q\cup \{\infty \}}$ and $T$ is an elliptic mapping of order $k$ in the normalizer of $\Gamma_{0}(N)$.

ON SUBORBITAL GRAPHS FOR THE MODULAR GROUP

This paper proves a conjecture of G. A. Jones, D. Singerman and K. Wicks, that a suborbital graph for the modular group is a forest if and only if it contains no triangles.