Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.


Introduction
e Hecke group, H(λ), introduced by Hecke in [1], is the group generated by the two Möbius transformations where λ is a real number such that λ � λ q � 2 cos(π/q) and q is an integer greater than 2 or λ > 2. When q � 3, H(λ 3 ) � H(1) is the modular group PSL(2, Z). If q � 4, 6, it is known, see [2], that H(λ q ) � H( �� m √ ), where m � 2, 3 consists of all transformations of the following two types: However, Rosen [3] showed that the above two transformations need not to be in H(λ) if λ ≠ 1, where r 0 is an integer and r i is a positive integer for i � 1, . . . , n. Later, Keskin [4] presented the Fuchsian group H( �� m √ ) for a squarefree positive integer m, which consists of all mappings of the forms (2) and (3).
In 1991, Jones et al. [5] studied the modular group H(1) by applying the idea of the suborbital graphs for a permutation group, introduced in [6]. Later, Akbas [7] proved his conjecture stating that a suborbital graph for the modular group is a forest if and only if it contains no triangles. ey completely characterized circuits in suborbital graphs for the modular group.
ese studies lead us to explore infinite paths in suborbital graphs for the Fuchsian group H( �� m √ ). In [8], Yayenie  ) where m is a prime number. We divide this work into four sections. In Section 2, we will show the way to construct a suborbital graph G(∞, ) and we give the edge conditions for two vertices that are joined in the graph. In Sections 3 and 4, we investigate vertices on the infinite path of minimal length in the graph G(∞, (u/v) �� m √ ) for two cases: (m, v) � 1 and (m, v) � m, respectively; here, ordered pairs denote greatest common divisors. Finally, we represent each vertex on the infinite path of minimal length by recurrence relations and determine the limit point of the sequence of the vertices.

Suborbital Graphs for
Q. e following remark was given by Yayanie in [8] Q. We will give a construction of suborbital graphs for the Fuchsian group Q. e orbits of this action are called suborbitals of H( �� m √ ). e suborbital containing (α, β) is denoted by O(α, β). We can form a suborbital graph G(α, β) whose vertices are the elements of �� m √ Q, and there is In the latter case, G(β, α) is just G(α, β) with reversed arrows and we call G(β, α) and G(α, β) paired suborbital graphs. In the case G(β, α) � G(α, β), the graph consists of pairs of oppositely directed edges, and we replace each pair with an undirected edge for convenience. We call the graph self-paired. Since Q. e following two theorems are valid for prime number m and we can use the same technique in the proofs of eorem 1 and 2 in [9], which were stated for m � 2 and 3.
) if and only if ry − sx � ± v and either.
(i) x ≡ ± ur mod v, y ≡ ± us mod v and m | s or (ii) x ≡ ± mur mod v, y ≡ ± mus mod v and m | y.
if and only if either.
Here, the choice of signs for x and y are always the same. By using eorems 1 and 2, we obtain the following two corollaries that characterize a self-paired graph. en the suborbital graph G(∞,

Corollary 2. Let u and v be relatively prime and m prime such that
Next, we will show the existence of an integer k such that mu 2 + kmu + 1 ≡ 0 mod v.

Lemma 1. Let u and v be relatively prime and m prime such
, it is seen that mu 2 + kmu + 1 ≡ 0 mod v is satisfied. Note that k and l are uniquely determined.

Infinite Path of Minimal Length in
a path and an infinite path, respectively.
and v i+1 must be the farthest vertex which can be joined with the vertex v i .
In this section, we focus on the infinite path of minimal length in the suborbital graph G(∞,

respectively. No nearest vertex exists. (ii) e farthest vertices which can be joined with
respectively. No nearest vertex exists. (iii) e farthest vertices which can be joined with ( respectively. No nearest vertex exists. Proof. (i) For the right side of (u/v) �� m √ , we assume that there exists an edge With this and the fact that uy < vx, we can replace where t/s is in Q + . Let d be the greatest common divisor of su + t and sv; then, we get (su + t/d, sv/d) eorem 1 gives the conditions when this edge exists. Since (m, v) � 1, we have m∤v so case (i) in eorem 1 cannot happen. en, we thus consider case (ii). In this case, We will find the largest value of t/s by defining a function f : , which is negative for every nonnegative z. is implies that the maximum occurs at z � 0 and maximum value is By eorem 1, it now suffices to show that (kmu + 1)/kmv is an irreducible fraction. us, ) and is the farthest one joined with (u/v) �� m √ . We also see that is implies that there is no such nearest point joined International Journal of Mathematics and Mathematical Sciences 3 e proof is similar to the previous case. Next, we will consider the left side of (u where p/q is in Q + . Let c be the greatest common divisor of qu − p and qv; then, we get ((qu − p)/c, qv/c) � 1 and By eorem 1, case (i) cannot happen. So, we will consider case (ii). en, we have (qu − p)/c ≡ ± mu 2 We define a function f : , which is positive for every nonnegative z.
is implies that the minimum occurs at z � 0 and minimum value is By eorem 1, it now suffices to show that (lmu − 1)/lmv is an irreducible fraction. us, ) and is the farthest one joined with (u/v) �� m √ . We also see that is implies that there is no such nearest point joined is case is done by using a similar argument to that of the previous case. □ Corollary 3. Let u and v be relatively prime and m prime such that (m, v) � 1. If there are integers k, l such that mu 2 + kmu + 1 ≡ 0 mod v and mu 2 ) are the farthest vertices which can be joined

Corollary 4. Let u and v be relatively prime and m prime. If
whose vertices are in the set (31)

Corollary 5. Let u and v be relatively prime and m prime. If
then there is an infinite path of minimal length: 4 International Journal of Mathematics and Mathematical Sciences whose vertices are in the set

Infinite Path of Minimal Length in
In the previous section, we provided the existence of infinite path of minimal length in the suborbital graph We find that the existence property is also valid for the suborbital graph (i) e farthest vertices which can be joined with on the right and the left are

respectively. No nearest vertex exists. (ii) e farthest vertices which can be joined with
respectively. No nearest vertex exists.

Corollary 6. Let u and v be relatively prime and m prime
such that (m, v) � m. If there are integers k, l such that u 2 + ku + 1 ≡ 0 mod v and u 2 − lu + 1 ≡ 0 mod v, then

Corollary 7. Let u and v be relatively prime and m prime. If
whose vertices are in the set (39)

Corollary 8. Let u and v be relatively prime and m prime. If
then there is an infinite path of minimal length: whose vertices are in the set International Journal of Mathematics and Mathematical Sciences 5

Continued Fractions and Recurrence Relations
From results in Sections 3 and 4, we have that any vertex on the infinite path of minimal length can be represented by a continued fraction expansion. As a continued fraction is related to recurrence relations, we use them to investigate vertices on the infinite path of minimal length. We conclude this section by finding the limit point of the sequence of the vertices. Let a n n∈N , b n n∈N 0 be sequences of complex numbers with a n ≠ 0 for n ≥ 1 and t m n∈N 0 , T m n∈N 0 be sequences of Möbius transformations defined as follows: t n (z) � a n b n + z , for n ∈ N, We consider T 0 (0), T 1 (0), T 2 (0), T 3 (0) and so on and form a continued fraction of the form For convenience, we denote this by In [10], the n th numerator A n and the n th denominator B n of a continued fraction as in (43)  with initial conditions For a given sequence z n n∈N 0 , T n (z n ) can be written as T n z n � A n + A n− 1 z n B n + B n− 1 z n , n � 0, 1, 2, . . . , and then Now we consider infinite paths in suborbital graph ). For the case when (m, v) � 1, the infinite path of minimal length for the right direction in Corollary 4 gives a n � − (1/m) and b n � − k for n ≥ 1. By recurrence relations in (45), we obtain B n � − mA n+1 , and then, we have a vertex on this path: Similarly, we have a vertex on the infinite path of minimal length for the left direction is Theorem 7. If (m, v) � 1 and k ≥ 2, then we have Proof. From the recurrence relation, we have with A − 1 � 1 and A 0 � 0. e characteristic equation for the relation (53) is which gives two roots 2m , en, any solution of (9) have the form By using the initial conditions, we have A 0 � α + β � 0, which implies As k ≥ 2 and |f| ≤ 1, we get f � (mk− ��������� m 2 k 2 − 4m √ )/2m. erefore, we obtain that the sequence of the vertices of infinite path of minimal length (30) converges to We observe that the limit points in Corollaries 23 and 24 are not in the set �� m √ Q, but the limit points in Corollaries 25 and 26 will be in the set 2 if k � l � 2.

Data Availability
ere are no data for supporting this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.