Suborbital graphs for the group Γ 2

Transcription

1 Hacettepe Jounal of Mathematics and Statistics Volume , Subobital gaphs fo the goup Γ 2 Bahadı Özgü Güle, Muat Beşenk, Yavuz Kesicioğlu, Ali Hikmet Değe Keywods: Abstact In this pape, we investigate subobital gaphs fomed by the action of Γ 2 which is the goup geneated by the second powes of the elements of the modula goup Γ on ˆQ. Fistly, conditions fo being an edge, self-paied and paied gaphs ae povided, then we give necessay and sufficient conditions fo the subobital gaphs to contain a cicuit and to be a foest. Finally, we examine the connectivity of the subgaph F u, and show that it is connected if and only if 2. Modula goup, Goup action, Subobital gaphs 2000 AMS Classification: 20H05, 05C25 Received 13/11/2013 : Accepted 27/05/2014 Doi : /HJMS Intoduction Let PSL2,R denote the goup of all linea factional tansfomations T : z az + b, whee a, b, c and d ae eal and ad bc = 1. cz + d In tems of matix epesentation, the elements of PSL2,R coespond to the matices a b ± ; a, b, c, d R and ad bc = 1. This is the automophism goup of the uppe half plane H := {z C : Imz > 0}. The modula goup Γ=PSL2, Z, is the subgoup of PSL2,R such that a, b, c and d ae integes. It is geneated by the matices U = ; V = Dept. of Math., Kaadeniz Tech. Uni., Tukey, bogule@ktu.edu.t Coesponding Autho. Dept. of Math., Kaadeniz Tech. Uni., Tukey, mbesenk@ktu.edu.t Dept. of Math., Recep Tayyip Edogan Uni., Tukey, yavuzkesicioglu@yahoo.com Dept. of Math., Kaadeniz Tech. Uni., Tukey, ahikmetd@ktu.edu.t

2 1034 with defining elationships U 2 = V 3 = I, whee I is the identity matix. Γ is a Fuchsian goup of signatue 0; 2, 3,, so it is isomophic to a fee poduct C 2 C 3. Define Γ m as the subgoup of Γ geneated by the m th powes of all elements of Γ. Especially, Γ 2 and Γ 3 have been studied extensively by ewman [13,14,15]. It tuns out that, { } Γ 2 a b = Γ : ab + bc + cd 0 mod 2, by Rankin [Eq , 16]. Fom the equation ab + bc + cd 0 mod 2, we see that at least one of the integes a, b, c, d must be even. Suppose fist that a = 2a 0. Then using the deteminant, we have that b and c ae odd. So, d must be odd as well. Hence, we 2a b get the elements of Γ 2 as the matices a b get the elements of the fom c 2d c will be even. To sum up, Γ 2 has thee types of elements 2a b a 2b a b,, 2 c 2d. Similaly, supposing d = 2d 0, we can. Lastly, if a o d is not even, then both b and whee b, c and d of the fist, a and d of the second and a, b, c of the thid matix ae odd Theoem. [13] The goup Γ 2 is the fee poduct of two cyclic goups of ode 3, and Γ : Γ 2 = 2, Γ = Γ Γ The elements of Γ 2 may be chaacteized by the equiement that the sum of the exponents 0 1 of be divisible by The idea of a subobital gaph has been used mainly by finite goup theoists. In [7], Jones, Singeman and Wicks showed that this idea is also useful in the study of the modula goup, whee they poved that the well-known Faey Gaph is an example of a subobital gaph. Futhemoe, they poved the following esult: Theoem A. The subobital gaph G u,n of Γ contains diected tiangles if and only if u 2 ± u mod n. Moeve they posed the conjectue: G u,n is a foest if and only if it contains no tiangles, that is, if and only if u 2 ± u mod n. Akbas poved in [2] that this conjectue is tue. By simila aguments, we concen with subobital gaphs of Picad goup P, which is the subgoup of PSL2, C with enties coming fom Z[i] in [3]. Since Z[i] is a unique factoization domain with finitely many units, ou expectation was to find simila fomulas. Consequently, theoem A was impoved as Theoem B. The subobital gaph G u, of P contains diected tiangles if and only if ε 2 u 2 εu ± 1 0 mod. In this study, we will continue to investigate the combinatoial popeties of these gaphs fo the goup Γ 2. It is an impotant subgoup of Γ since all the goups Γ m can be expessed in the tems of Γ, Γ 2, Γ 3. The pupose of this pape is to chaacteize all cicuits in the subobital gaph and connectedness fo Γ 2. As it can be seen fom Section 3, we show that the main diffeence is in connectedness of elated gaphs.

3 The action of Γ 2 on ˆQ Evey element of ˆQ can be epesented as a educed faction x with x, y Z and y x, y = 1. This epesentation is not unique, because x = x. We epesent as y y 1 a b = 1. The action of the matix on x 0 0 i a b : x ax + by y cx + dy. Hence, the actions of a matix on x x and on ae identical. If the deteminant of the y y a b matix is 1 and x, y = 1, then ax + by, cx + dy = Tansitive Action Lemma. i The action of Γ 2 on ˆQ is tansitive. ii The stabilize of a point is an infinite cyclic goup. Poof. i Hee we only pove the case that any element of the fom a of ˆQ is sent 2b by an element of Γ 2. The est ae simila. Let a ˆQ, a, 2b = 1. Thee exist integes 2b x 0 and y 0 such that ay 0 2bx 0 = 1 known as Bezout s identity [8]. Hence, we have a x0 that T := Γ. All solutions of the equation ay 2bx = 1 ae x = x 2b y 0 + an 0, y = y 0 + 2bn fo n Z. If x 0 is odd, x would be even by taking n-odd. So, x 0 can be chosen as an even numbe. Hence, T Γ 2 and T = a means that a is in the obit 2b 2b of. ii By i, since the stabilizes of any two points in ˆQ ae conjugate in Γ 2, it is 1 2 sufficient to conside the stabilize Γ 2 of. It is clea that Γ 2 =. 0 1 We emak that Lemma 2.1 i can be poven by using the signatue of Γ 2 as well. Thee is a homomophism θ : Γ C 2 = {e, α} defined by θu = α, and θv = e. The kenel is Γ 2. By the pemutation theoem [19], Γ 2 has signatue 0; 3, 3,. It means that thee is only one obit, so the action is tansitive Impimitive Action. We now discuss the impimitivity of the action of Γ 2 on ˆQ. Fo this, let G, Ω be a tansitive pemutation goup, consisting of a goup G acting on a set Ω tansitively. An equivalence elation on Ω is called G-invaiant if, wheneve α, β Ω satisfy α β, then gα gβ fo all g G. The equivalence classes ae called blocks. We call G, Ω impimitive if Ω admits some G-invaiant equivalence elation diffeent fom i the identity elation, α β if and only if α = β; ii the univesal elation, α β fo all α, β Ω. Othewise, G, Ω is called pimitive. These two elations ae supposed to be tivial elations Lemma. [4] Let G, Ω be a tansitive pemutation goup. G, Ω is pimitive if and only if G α, the stabilize of α Ω, is a maximal subgoup of G fo each α Ω. Fom the above lemma we see that wheneve, fo some α, G α H G, then Ω admits some G-invaiant equivalence elation othe than the tivial one and the univesal one.

4 1036 Because of the tansitivity, evey element of Ω has the fom gα fo some g G. Thus one of the non-tivial G-invaiant equivalence elations on Ω by H is given as follows: gα g α if and only if g gh. The numbe of blocks equivalence classes is the index G : H and the block containing α is just the obit Hα. Let and let Γ 2 0 be defined by { } Γ 2 a b 0 := Γ 2 : c 0 mod. Then Γ 2 0 is a subgoup of Γ 2. It is clea that Γ 2 Γ 2 0 Γ 2 fo and Γ 2 Γ 2 0 Γ 2 fo > Lemma. Γ 0 : Γ 2 0 = 2. In fact, 1 0 Γ 2 0 Γ 2 1 0, Γ 0 = Γ 2 0 Γ 2 1 0, is odd is even Poof. Fist, we suppose that is even. Let s show that Γ a b Γ = Γ 0. We have that T := Γ c d 0 with ad bc = 1. Hee, a and d ae odd. If b is even, T would be an element of Γ 2 0. We x y suppose that b is odd. Hence, it can be witten as T =. 1 c z 1 1 a b x y Then, we have that =. Let s say that + 1 c d c z a + c b + d x y =. a + c + 1 b + d + 1 c z }{{} A As b + d is even, A Γ 2 0. ow, let be odd. In this case, assume that b and c ae even in T. Then a and d ae odd. Hence, T is an element of Γ 2 0. Moeove, it can be witten as T = x y c z. As above, let s say that a b } c a {{ d b } B x y. Since d b is even, B Γ 2 c z 0. In the case: b-even and c-odd, it is clea that B Γ 2 0. If a and d ae even in the equation ad bc = 1, B Γ 2 0 again. Finally if a is odd and d is even o vice vesa, the esult is the same. Consequently, we obtain that Γ 0 : Γ 2 0 = 2. Theefoe, fom the above constucted equivalence elation ", we get Γ 2 -invaiant equivalence elation on ˆQ by Γ 2 0. It is clea that, by Lemma 2.3, Γ 2 acts impimitively on ˆQ. Let v = s and w = x y be elements of ˆQ. Because of the tansitive action, we have that v = g 1 and w = g 2 fo some elements g 1, g 2 Γ 2 of the fom =

5 g 1 := s x, g 2 := y Fom the elation we get v w if and only if g 1 1 g2 Γ2 0, v w if and only if y sx 0 mod. By ou geneal discussion of impimitivity, the numbe of blocks unde is given by Ψ = Γ 2 : Γ 2 0. So the block of is obtained as [ ] := { } x y ˆQ y 0 mod Lemma. Ψ = p dividing whee the poduct is ove the distinct pimes p p Poof. To calculate Ψ we use two decomposition of the index Γ : Γ 2 0 as the following Γ : Γ 2 Γ 2 : Γ 2 0 = Γ : Γ 0 Γ 0 : Γ 2 0. Hee, Γ : Γ 2 = 2 and Γ : Γ 0 = p ae well-known by [13,16] and p [16,17] espectively. We pove that the index of Γ 0 : Γ 2 0 is equal to 2 in Lemma 2.3. Witing these values in above equation, the esult is obvious. 3. Subobital Gaphs fo Γ 2 on ˆQ In[18], Sims intoduced the idea of the subobital gaphs of a pemutation goup G acting on a set, these ae gaphs with vetex-set, on which G induces automophisms. We summaise Sims theoy as follows: Let G, be tansitive pemutation goup. Then G acts on by gα, β = gα, gβg G, α, β. The obits of this action ae called subobitals of G. The obit containing α, β is denoted by Oα, β. Fom Oα, β we can fom a subobital gaph Gα, β : its vetices ae the elements of, and thee is a diected edge fom γ to δ if γ, δ Oα, β. A diected edge fom γ to δ is denoted by γ δ. If γ, δ Oα, β, then we will say that thee exists an edge γ δ in Gα, β. In this pape ou calculation concens Γ 2, so we can daw this edge as a hypebolic geodesic in the uppe half-plane H, that is, as euclidean semi-cicles o half-lines pependicula to the eal line. The obit Oβ, α is also a subobital gaph and it is eithe equal to o disjoint fom Oα, β. In the latte case Gβ, α is just Gα, β with the aows evesed and we call, in this case, Gα, β and Gβ, α paied subobital gaphs. In the fome case Gα, β = Gβ, α and the gaph consists of pais of oppositely diected edges; it is convenient to eplace each such pai by a single undiected edge, so that we have an undiected gaph which we call self paied Definition. By a diected cicuit in a gaph we mean a sequence v 1, v 2,..., v m of diffeent vetices such that v 1 v 2... v m v 1, whee m 3. If m = 3, then the cicuit, diected o not, is called a tiangle. If m = 2, then we will say the configuation v 1 v 2 v 1 is self paied.

6 1038 A gaph which contains no cicuit is called a foest. The above ideas ae also descibed in a pape by eumann [12] and in books by Tsuzuku [20] and by Biggs and White [4], the emphasis being on applications to finite goups. The eade is efeeed to [1, 2, 3, 6, 7, 9, 10, 11] fo some elevant pevious wok on subobital gaphs. If α = β, then Oα, α = {γ, γ γ } is the diagonal of. The coesponding subobital gaph Gα, α, called the tivial subobital gaph, is self-paied: it consists of a loop based at each vetex γ. We shall be mainly inteested in the emaining non-tivial subobital gaphs. Since Γ 2 acts tansitively on ˆQ, each subobital contains a pai, v fo some v ˆQ; witing v = u, u, = 1 and 0. We denote this subobital by Ou, and the coesponding subobital gaph by G u, Gaph G u,. If v =, we would have the simplest subobital gaph, namely G 1,0 = G 1,0. Theefoe, we can take v Q. Let v = u Q. The necessay and sufficient condition fo O, v = O, v is that v and v ae in the same obit of Γ 2. Since Γ 2 is geneated by z : v v + 2, then z u = u+2 = u. Theefoe, we have that = and u u mod 2. Hence, G u, = G u, if and only if = and u u mod 2. Consequently, fo a fixed thee ae 2ϕ distinct subobital gaphs G u, whee ϕ is Eule s phi function Theoem. x Gu, if and only if i If is even, then x ±u mod, y ±us mod, y ±us mod 2 and y sx =. ii If s is even, then x ±u mod 2, y ±us mod and y sx =. iii If and s ae odd, then x ±u mod, y ±us mod 2 and y sx =. Poof. i Let be even. By the tansitivity of Γ 2, without loss of geneality, we assume that < x whee all lettes ae positive integes. Thus, we have that y sx < 0. Since a b x Gu,, thee exist some T = Γ 2 such that T 1, u 0 =, x. a b 1 u x A sx < 0, the multiplication of is equal to o 0 x. If the fist case is valid, we have that a =, c = s, au + b = x and s y cu + d = y. That is, x u mod and y us mod. Since is even, then a is also even. To have T Γ 2, d must be odd. Fom us + d = y, we have that y ±us mod 2. ii Suppose s is even. In a simila way, we see that b and c must be even because T 1 0 = a b 1 u x = a. As in i, we may assume that =. s c 0 Hence, we have that a =, c = s, au+b = x, cu+d = y and y sx =. That is, u + b = x and us + d = y. Since b is even, we have that x u mod 2 and y us mod. iii Let and s be odd. With simila agument, it can be seen that d must be even. Fom the same matix equation in ii, we obtain that x u mod and y us mod 2.

7 1039 In the opposite diection, we shall pove i fo minus sign. Suppose that is even, x u mod, y us mod, y us mod 2 and y sx =. In this case, thee exist integes b, d such that x = u b, y = us d. So, it is b clea that Γ 2 which means s d x Gu,. Because = y sx = us d s u b. This implies d + sb = 1. As is even, d must be even. Othewise, it contadicts ou hypothesis. With simila agument, we obtain the elements 2b b of Γ 2 which ae and fo ii and iii espectively. s d s 2d 3.3. Theoem. All subobital gaphs fo Γ 2 on ˆQ ae paied. Poof. Because of the tansitivity of Γ 2, it is sufficient to show that G, u G u,. It means that thee is no T Γ 2 which sends the pai, u to the pai u,. On the contay, assume that T = u and T u =. By compaing the deteminants, we have that a b 1 u u 1 a b 1 u u 1 = o = In the fist case, we obtain a = u, c =, au + b = 1 and cu + d = 0. That u b is, d = u and u 2 = 1 + b. Taking T = we see that the only case fo u T to be an element of Γ 2 is that and b must be even. Since u 2 = 1 + b, then u 2 1 mod b. As and b ae even, u 2 1 mod 4 which has no solution. Fo u b the second case, taking T =, simila contadiction is obtained. u 3.4. Coollay. Thee ae no self-paied subobital gaphs fo Γ 2 on ˆQ.. In section 2 we intoduced, fo each intege, a Γ 2 -invaiant equivalence elation on ˆQ, with x if and only if y sx 0 mod. If x in Gu,, then Theoem s y 3.2 implies that y sx = ±, so x. Thus, each connected component of Gu, lies s y in a single block fo, of which thee ae Ψ, so we have: 3.5. Coollay. The gaph G u, is a disjoint union of Ψ subgaphs Subgaph F u,. We epesent the subgaph of G u, whose vetices fom the block [ ] = {x/y ˆQ y 0 mod } by F u, Coollay. The gaph G u, consists of Ψ disjoint copies of F u,. Poof. The vetices of each subgaph fom a single block with espect to the Γ 2 -invaiant equivalence elation defined by y sx 0 mod. Theefoe, if x 1 x 2 is an edge in the subgaph F u,, T x 1 T x 2 is also an edge in any othe subgaph with T Γ 2 because of the tansitivity of Γ 2 on ˆQ. ow, Theoem 3.2 immediately gives: 3.7. Theoem. x Fu, if and only if i If is even, then x ±u mod, y ±us mod, y ±us mod 2 and y sx =. ii If s is even, then x ±u mod 2, y ±us mod and y sx =. iii If and s ae odd, then x ±u mod, y ±us mod 2 and y sx =.

8 Theoem. Γ 2 0 pemutes the vetices and the edges of F u, tansitively. Poof. Let v, w be any vetices of F u,. Since Γ 2 acts on ˆQ tansitively, thee exist a b T Γ 2 such that T v = w. Taking T =, v = k 1 and w = k 2 l 1 l 2 we see that c. It means that Γ 2 0 pemutes the vetices of F u,. Let x e 1 1 y 1 b1 and x e 2 2 y 2 b2 be any edges of F u,. We can give following diagam: 1, u T2 0 x2, y 2 b2 T 1 x1 y 1, b1 T 2 T 1 1 x1 x2 By this epesentation, we have T 1 = and T y 1 2 =. Since y 2 T 2 T 1 1 has the fom fo some intege k, then T := T k 2 T 1 1 Γ 2 0. It is clea that T x1 y 1 = x 2 y 2 and T b1 = b2. Since T is an element of a goup of hypebolic isometies of H, geodesics ae sent to geodesics unde its action. So, T tansfom the edges e 1 to e 2. Consequently, Γ 2 0 pemutes the edges of F u, Lemma. Thee is an isomophism F u, F u, given by v v. Poof. It is clea that v v is one-to-one and onto. Let s show that the stuctue is peseved. Hee, it means that if a b F u,, then a b F u,. Suppose that x Fu, and is even. By Theoem 3.7i, taking < x, we have that x u mod, y us mod, y us mod 2 and y sx =. Since < x, then > x. Taking x u mod, y us mod, y us mod 2 and y + sx =, we have that x F u,. Fo othe conditions, the est ae simila Lemma. If M, then thee is a homomophism F u, F u,m given by v v/m., s < x y x Poof. We suppose that ae adjacent vetices in Fu, and < x and y that is witten as Fu,. If is even, then x u mod, y s us mod, y us mod 2 and y sx = 1. Since M, x u mod M, ym usm mod M, ym us mod 2M. y sx = 1 is also tue fo M. Fo othe conditions, the est ae simila Theoem. F u, contains diected tiangles if and only if u 2 u+1 0 mod. Poof. Suppose that F u, contains a diected tiangle. Because of the tansitive action, the fom of diected tiangle can be taken as u < fo some intege. Fist, let u be even. Fom the second edge, we have u = 1 and u 2 mod by Theoem 3.2. So, we obtain u 2 + u mod. Similaly, if u >, then we see that u 2 u mod. ow, is even. By applying Theoem 3.2 to the second edge, we have u = 1 and u 2 mod 2, giving u 2 + u mod 2. It is impossible, because thee is no solution fo this equivalence. Finally, suppose that u, ae odd. Again, fom the second edge, we have u = 1 and u 2 mod, giving u 2 + u mod. If u >, it would be u2 u mod. Combining all of the equivalences, we obtain u 2 u mod. Convesely, if u 2 u mod, we see that eithe u + 1 u 2 mod o u + 1 u 2 mod. Theoem 3.2. implies that thee is an edge u u+1 with

9 1041 u < u+1 in Fu, o u u 1 with u > u+1 in Fu,. tiangle u u±1 in Fu,. Consequently, thee is a diected 1 Let us give some examples. Fo u, -odd, o is a diected tiangle in F3,13. Fo u-even and -odd, o is a diected tiangle in F2,7. Fo -even, we know that thee is no tiangle. Obsevation. We know that thee is no tiangle in F u,20 fo -even by Theoem Because of the elationships between elliptic elements with cicuits, ou expectation is u 2b that thee is no elliptic element of ode 3 of the fom Γ 2. Indeed, being 2 0 d an elliptic element of ode 3, it is well-known that u + d = ±1. Taking deteminant 1 d 2b of, we have d d 2 4b 2 0 d 0 = 1. It is clea that thee is no solution fo d d 2 1 mod 4. On the othe hand, we know that the subobital gaph fo modula goup is a foest if and only if it contains no tiangles [2]. Using this fact, we can give the following esult; Coollay. The gaph G u, is a foest if and only if u 2 ± u mod Connectedness. In this last section, we examine the connectedness of F u, Definition. A subgaph K of G u, is called connected if any pai of its vetices can be joined by a path in K Theoem. The subgaphs F 0,1 and F 1,1 ae connected. Poof. Hee, to see the situation bette, we wite the edge conditions fo F 0,1 and F 1,1 by Theoem 3.2 explicitly. Case F 0,1: x F0,1 if and only if i If -even, then y-odd and y sx = 1. ii If s-even, then x-even and y sx = 1. iii If, s-odd, then y-even and y sx = 1. We will show that each vetex a of F0,1 can be joined to by a path in F0,1. It is clea b fo b = 1. Since a, b = 1, we can wite the equation ad bc = 1 by Bezout s identity. Fo this pai c, d satisfying the equation we claim that a can be joined with c by above b d edge condition. Subcase1. Suppose a-even. By the equation we have that b, c must be odd and thee ae two possibilities fo d. If d-odd, then a i c means that we have c a by i. If b d d b d-even, then c ii a. d b Subcase2. Let b-even. By the equation we have that a, d must be odd and thee ae two possibilities fo c. If c-odd, then c iii a. If d-even, then a ii c. d b b d Subcase3. Assume that a-odd and b-odd. By the equation it is impossible that c, d ae odd o even at once, so thee ae two possibilities. If c-odd and d-even, then a iii c. If b d c-even and d-odd, then c i a. d b Consequently F 0,1 is connected. Case F 1,1: x F1,1 if and only if i If -even, then y-even and y sx = 1.

10 1042 ii If s-even, then x-odd and y sx = 1. iii If, s-odd, then y-odd and y sx = 1. Taking a vetex a in F1,1, thee exists the equation ad bc = 1 by Bezout s identity. b We shall show that a is adjacent to vetex c in F1,1. b d Subcase1. Suppose a-even. By the equation we have that b, c must be odd and thee ae two possibilities fo d. If d-odd, then c iii a. If d-even, then a i c. d b b d Subcase2. Let b-even. By the equation we have that a, d must be odd and thee ae two possibilities fo c. If c-odd, then a ii c. If c-even, then c i a. b d d b Subcase3. Assume that a-odd and b-odd. By the equation it is impossible that c, d ae odd o even at once, so thee ae two possibilities. If c-odd and d-even, then c ii a. If d b c-even and d-odd, then a iii c. b d Consequently, F 1,1 is connected Theoem. The subgaphs F 1,2 and F 3,2 ae connected. Poof. We shall show that each vetex v = a b 1 of F1,2 is joined to by a path 2b in F 1,2. Since the patten is peiodic with peiod 2, we can show by induction on b. If b = 1, then v = a can be joined with. If a = 1, it is clea that 1 1. If a = 1, then because 1 3 mod 4 and = 2. If a = 5, then 1 5. The same holds fo the est peiodically. So we can assume that b 2. To complete the poof, we show that v is adjacent to a vetex w = a 2b 1 with b 1 < b. It means that, w is connected by a path to, and hence so is v. As a, b = 1, thee exist integes c, d such that ad bc = 1. Fo some k Z, eplacing c and d by c + ka and d + kb, we can suppose 0 < d < b. i If c is odd, then w = c can be joined with a. Indeed, a > c gives that 2d 2b 2b 2d a 2d c 2b = 2 and c a mod 4. If c a mod 4, taking c a mod 4 we obtain a < c by 2bc 2ad = 2. Hence, if c is odd, a c is adjacent to in F1,2. 2b 2d 2b 2d ii If c is even, then a c is odd. As 0 < b d < b, we can take w = a c, 2b d adjacent to a because 2bc cd = 2. Hee, if 2a c 0 mod 4, then we have that 2b a c a mod 4 and 2ad bc = 2. Consequently, F 1,2 is connected. By the isomophism F 1,2 F 1,2 = F v 3,2, F 3,2 is v also connected Coollay. All gaphs F u,2 ae connected Coollay. The gaph G u,2 has 2 ψ2 = 6 connected components. Its blocks ae [ ], [1], [0]. The connected components of [ ] ae F 1,2 and F 3, Theoem. The subgaphs F 1,3, F 2,3, F 4,3 and F 5,3 ae not connected. Poof. It is sufficient to study with F 1,3 and F 2,3. Because thee is an isomophism fom F 1,3F 2,3 to F 5,3F 4,3 espectively. Case F 1,3: If F 1,3 is connected, then each vetex v = a would be joined to. We shall 3b show that no vetices of F 1,3 whee 1 < v < 2 ae adjacent to. Futhe, we asset that thee is no such a vetex v adjacent to vetices outside this inteval. Of couse, thee is at least some vetex of F 1,3 in this stip. Suppose 2 c < 1 < a < 2. Then we have 3 3d 3b c < 3 < a. This is impossible because cd ad = 1. Similaly, if 1 < k < f 7, then d b 3l 3e 3 k < 4 < f contadicts ke lf = 1. It means that no vetices of F1,3 between 1 and 2 l e ae adjacent to and that F 1,3 is not connected.

11 1043 Figue 1. The subgaph F 1,3 Case F 2,3: As above, let s show that no vetices of F 2,3 between 3 and 2 ae adjacent to 2 vetices outside this inteval. Suppose that 1 x < 3 < a < 2 and x < a F2,3. 3y 2 3b 3y 3b Then we have that x < 9 < a and xb ay = 1. By [7], we obtain that x = 4, y = 1, y 2 b a = 5 and b = 1. But 4 5 is not in F2,3. If 2 < x < 2 < a < 8 and x < a F2,3, y 3b 3 3y 3b then we would have x < 6 < a and xb ay = 1. It is impossible because of well-known y b Faey sequence. Consequently, F 2,3 is not connected Coollay. All gaphs F u,3 ae not connected. Figue 2. The subgaph F 2, Theoem. The subgaphs F 1,4, F 3,4, F 5,4 and F 7,4 ae not connected. Poof. As emaked in the poof of Theoem 3.18, it is sufficient to study with F 1,4 and F 3,4. Case F 1,4: We will show that no vetices in F 1,3 between 1 and 1 ae adjacent to vetices 2 outside this inteval. Suppose 1 a < 1 < x < 1. Then we have a < 2 < x. This is 4 4b 2 4y b y

12 1044 impossible because ay bx = 1. Similaly, if a contadiction. So F 1,4 is not connected. a 4b < 1 < x 4y 7 4, then a b < 4 < x y < 7 is Case F 3,4: As above, it is seen that no vetices of F 3,4 between 1 and 2 ae adjacent to vetices outside this inteval. Consequently, F 3,4 is not connected Theoem. The subgaph F u, is connected if and only if 2. Poof. If F u, is connected, we know that 4 by [7]. Fo = 3, 4, we poved that F u, is not connected by Theoem 3.18 and Convesely, if 2, the esult immediately follows fom Theoem 3.14 and Refeences [1] Akbaş, M. and Başkan, T. Subobital gaphs fo the nomalize of Γ 0, T. J. of Mathematics, 20, , [2] Akbaş, M. On subobital gaphs fo the modula goup, Bull. London Math. Soc., 33, , [3] Beşenk, M. et al. Cicuit lengths of gaphs fo the Picad goup, J. Inequal. Appl., 2013:106, 8 pp., [4] Bigg,.L. and White, A.T. Pemutation goups and combinatoial stuctues, London Mathematical Society Lectue ote Seies, 33, CUP, Cambidge, 140 pp.,1979. [5] Dixon, J. D. and Motime, B. Pemutation Goups, Gaduate Texts in Mathematics 163, Spinge-Velag, [6] Güle, B.Ö. et al. Elliptic elements and cicuits in subobital gaphs, Hacet. J. Math. Stat., 40 2, , [7] Jones, G.A., Singeman, D. and Wicks, K. The modula goup and genealized Faey gaphs, London Mathematical Society Lectue ote Seies, 160, CUP, Cambidge, , [8] Jones, G.A. and Jones, J.M. Elementay umbe Theoy, Spinge Undegaduate Mathematics Seies, Spinge-Velag, [9] Kade, S., Güle, B. Ö. and Değe, A. H. Subobital gaphs fo a special subgoup of the nomalize, IJST. Tans A., 34 A4, , [10] Kade, S. and Güle, B. Ö. On subobital gaphs fo the extended Modula goup ˆΓ, Gaphs and Combinatoics, 29, no. 6, , [11] Keskin, R. Subobital gaphs fo the nomalize of Γ 0 m, Euopean J. Combin., 27, no. 2, , [12] eumann, P.M. Finite Pemutation Goups, Edge-Coloued Gaphs and Matices, Topics in Goup Theoy and Computation, Ed. M. P. J. Cuan, Academic Pess, [13] ewman, M. The Stuctue of some subgoups of the modula goup, Illinois J. Math., 6, , [14] ewman, M. Fee subgoups and nomal subgoups of the modula goup, Illinois J. Math., 8, , [15] ewman, M. Classification of nomal subgoups of the modula goup, Tansactions of the Ameican Math. Society, Vol.126, 2, , [16] Rankin, R. A. Modula Foms and Functions, Cambidge Univesity Pess, [17] Schoenebeg, B. Elliptic modula functions, Spinge Velag, [18] Sims, C.C. Gaphs and finite pemutation goups, Math. Z., 95, 76-86, [19] Singeman, D. Subgoups of Fuchsian goups and finite pemutation goups, Bull. London Math. Soc., 2, , [20] Tsuzuku, T. Finite Goups and Finite Geometies, Cambidge Univesity Pess, Cambidge, 1982.