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{\bf Refik Keskin and Bahar Demirt\"{u}rk}
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{\bf On Suborbital Graphs for the Normalizer of $\Gamma_{0}(N)$}
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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 $\Gamma_{0}(N)$.

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