The Cayley Property of Some Distant Graphs and Relationship with the Stern–Brocot Tree

One of the graphs associated with any ring R is its distant graph G(R,Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(R,\Delta )$$\end{document} with points of the projective line P(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}(R)$$\end{document} over R as vertices. We prove that the distant graph of any commutative, Artinian ring is a Cayley graph. The main result is the fact that G(Z,Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\mathbb Z,\Delta )$$\end{document} is a Cayley graph of a non-artinian commutative ring. We indicate two non-isomorphic subgroups of PSL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PSL_2(\mathbb Z)$$\end{document} corresponding to this graph.


Introduction
Let R be an associative ring with an identity. The automorphism group of the left R-module R 2 is the group GL 2 (R) of invertible 2x2 matrices with entries in R. The projective line over R is the orbit P(R) = R(1, 0) GL2(R) of the free cyclic submodule R(1, 0) under the action of GL 2 (R). The projective line P(R) is endowed with the symmetric relation Δ ("distant").
A pair (a, b) ∈ R 2 is called admissible if there exists (a , b ) ∈ R 2 (not necessary unique) such that where J = J(R) is the Jacobson radical of R. As an example of such a ring may serve any semi-local commutative one. Indeed, if {M i : i = 1, . . . , k} is the family of maximal ideals in R, then which easily follows from the Chinese Remainder Theorem. First we describe points of P(R) in terms of members of J and points of P(R). Let π : P(R) −→ P(R) denote the epimorphism induced by the canonical epimorphism R −→ R. The following is essentially proved in [5]. Obviously R(1, j) = R(1, j ) for j, j ∈ J, j = j . Further, if Observe thatρ is a bijection. Now consider an arbitrary section σ : where γ is an element of GL 2 (R) that carries σ(p) to R(1, 0): σ(p) γ = R(1, 0). Since γ(π −1 (p)) = π −1 (R(1, 0)) andρ is a bijection, ρ is a bijection on each of the cosets of π.
Since ρ is obviously surjective, D is so. Now D is bijective since ρ| π −1 (p) is bijective for every p ∈ P(R). Proof. Assume that (G, S) is an appropriate Cayley group with G written additively.
We will show that (J × G, J × S) is a Cayley group of R (here J is understood as the additive group of the Jacobson radical of R).
Let P(R) p −→ g p ∈ G be an appropriate bijection. Using Lemma 2.1 we get a bijection (being a composition of two bijections) Obviously we have for every p, q ∈ P(R), which finishes the proof. Theorem 2.3. If R is a ring satisfying (2.1) then its distant graph is a Cayley graph.
Proof. Assume that R λ∈Λ F λ , where F λ are skew-fields. The distant graph of F λ is a complete graph of order card(F λ )+1. Hence, using any group G λ of this order and S λ = G λ \ {1 λ }, where 1 λ is a neutral element of G λ , one obtains the distant graph as associated Cayley graph.
By our assumption we have It is not difficult to see that the distant graph of a product of rings is a tensor product of appropriate graphs. On the other hand, it is easy to see that one of the Cayley representations of a tensor product of Cayley graphs can be obtain by taking the product of the Cayley groups of the factors with the set of generators equal to the product of the appropriate sets of generators. Therefore the distant graph of P(R) has a Cayley group equal to Now we may use Corollary 2.2.
It is well known that every artinian commutative ring is a direct sum of local rings (see i.e. [1]). From this and Theorem 2.3 we have the following.

Examples
Example. Let n ≥ 1 be a natural number and consider the ring R = Z p n , where p is an arbitrary prime number. This is a local ring with a precisely one maximal ideal M = J(R) Z p n−1 . Therefore R Z p and the Cayley group of the distant graph of R is equal to Z p n−1 × Z p+1 with a generating set equal to Z p n−1 × (Z p+1 \ {0}).
Proof. It is a well known fact that for a simple, finite, connected, planar graph with v ≥ 3 vertices and e edges one has e ≤ 3v − 6. Assume that r = |R| ≥ 6. The degree of a vertex in a tensor product is equal to the product of degrees of its components. Since degree of each vertex is equal to r, we have: v ≥ r ≥ 6 and e = rv 2 . Therefore r 2 ≥ 3 > 3 − 6 v , which means that e > 3v − 6, and hence our graph is not planar. There is only one finite ring with unity of order 5, namely F 5 (here and in the sequel by F q we denote a finite field of order q ≥ 2). Its distant graph is the non-planar graph K 6 .

Corollary 3.2. Let R be a finite ring with unity. The distant graph of R is planar iff
Proof. If |R| = p where p = 2 or 3, then R = F p and its distant graph is the planar graph K p+1 , as is depicted on Fig. 1a, b.
If |R| = 4 then we have four rings to consider: Two former ones have the same planar graph depicted on Fig. 1c. Here we can use the proof of Theorem 2.3 to see that the Cayley group of the distant graph of Z 4 is equal to Z 2 × Z 3 with a generating set {(0, 1), (0, 2), (1, 1), (1, 2)}. The ring F 4 has the non-planar graph K 5 . The distant graph of the remaining ring F 2 × F 2 is equal to K 3 ⊗ K 3 . In this case we cannot use the necessary condition e ≤ 3v − 6 since for this graph we have e = 18 < 21 = 3v − 6. Nevertheless K 3 ⊗ K 3 has the graph K 5 as its minor. Indeed, let {u j , v j , w j : j = 1, 2, 3} be the set of vertices of K 3 ⊗ K 3 and let us decide that a i b j , where a, b ∈ {u, v, w} and i, j ∈ {1, 2, 3}, iff simultaneously a = b and i = j. The graph K 5 may be obtain as a minor of K 3 ⊗ K 3 after contracting the following edges: (u 1 , v 2 ), (u 2 , w 3 ), (u 3 , v 1 ) and  Fig. 2). Therefore K 3 ⊗ K 3 is not a planar graph in virtue of Wagner's theorem.
Example. Let R = T n (K) be the ring of lower n × n triangular matrices over the skew-field K. The Jacobson radical is The algebra R contains the subalgebra R = {A ∈ R : a 11 = a 22 = · · · = a nn }. It is a local ring and C = J(R ) ⊕ G is a Cayley representation of its distant graph.
From the proof of Theorem 2.3 it follows that S is equal to {(g 1 , . . . .g n ) : g j = 1 G } in the former case and G \ {1 G } in the latter one.
Example. This example shows that the condition (2.1) is sufficient but not necessary for R to have a distant graph with Cayley's property. Let R = M 2 (F q ), q ≥ 2. This is a non-commutative ring with no ideals. Nevertheless we show that it has a Cayley distant graph. It is known that: • the projective line of R has m = (q 2 + 1)(q 2 + q + 1) elements.
Moreover, from Result 7 and Theorem 1 of [6] it can be deduced that • the distant graph consists of q 2 + q + 1 disjoint maximal cliques, each of cardinality of q 2 + 1. The Cayley graph Γ(G, S) has the following disjoint maximal cliques: It may be checked by inspection that the Cayley graph of (G, S) satisfies the remaining properties described in the previous paragraph as well. Therefore (G, S) is a Cayley group of the distant graph of R. See Fig. 3, where a subgraph consisting of two disjoint maximal cliques is depicted in case q = 2.

The Distant Graph of the Ring of Integers
It can be shown that P GL(2, Z) is isomorphic to the automorphism group of the distant graph of the ring of integers. We omit the proof since it is not crucial for our considerations. On the other hand the group serving as a Cayley representation is a subgroup of the automorphism group. Therefore we will seek for such a representation in P SL(2, Z) < P GL(2, Z).
One can draw the distant graph G of Z using the Stern-Brocot tree ( [12]). In fact, to build G one has to perform the procedure twice. First starting with two points [1, 0] and [0, 1], then starting with [−1, 0] and [0, 1]. After obtaining two Stern-Brocot trees we identify their nodes [k, l] with vertices Z(k, l) of G, see Fig. 4. Recall that Z(k, l) belongs to P(Z) iff the greatest common divisor of k and l is 1.
In this section we will show that the distant graph of the ring of integers is a Cayley one. We find two non-isomorphic Cayley representations of this graph showing that such a representation is not an isomorphism invariant. In order to do this we use the approach developed by Hall in his proof of the Kurosh subgroup theorem [8].
as the reduced form of the element of G. This means that the identity is the void product; and for g = 1 each a i is an element = 1 of one of the free factors A ν , and no two consecutive terms a i , a i+1 (i = 1, . . . , t− 1) belong to the same free factor A ν . The length |g| of an element g is defined as zero for g = 1, and for g = 1 as the number t of terms in its reduced form. In fact Hall uses the so called semi-alphabetical ordering but in our setup we need only the partial ordering by length. Having a subgroup H < G let K = K(H) denote a set of all elements k ∈ H \ {1} such that k does not belong to the group generated by the elements of H which precede k in the semi-alphabetical ordering.
Consider two independent generators : n ≥ 0}. Now we show that both sets K i satisfy the following property (P): An arbitrary finite product of elements of K i has length greater than each of its factor.
Proof. We have to split calculations into separate cases (in both we will write if n j = 0. Therefore it is enough to show (P) with the assumption n j > 0. To end this note that a and b we denote a ∧ b = min(a, b).) From this it follows the general formula

(Having two numbers
(n j ∧ n j+1 ). Now (P) follows from the fact that |K 2n | = 4n − 1 and (4.1) Case i = 1. Similar considerations show that in this case Now (P) follows from |K 2n | = 4n + 1 and (4.1). Now it is obvious that K 2n , n > 1, does not belong to the subgroup generated by {K 2m : 0 ≤ m < n}. From this and the property (P) of K i we conclude that Now K 0 consists of involutions itself, and K 1 contains only involutions with the exception of one element K (1) 0 that is of order 3. Denote C i := K i , i = 0, 1. From Hall's proof of the Kurosh subgroup theorem it follows that As a byproduct of the proof of the property (P) we get that Now we will replace the sets of generators K i by other sets of generators S i in such a way that Γ(C i , S i ) become Cayley representations of G.
For n ≥ 0 put Proof. To show the left hand side equality in the even case we use induction on n. For n = 0 this follows from the definitions. Assume that this formula holds for some n ≥ 0. Then in the case i = 0 we have For i = 1 we have The odd case follows from the definitions and just proved equalities.
We intend to show that S In order to do this we define two involutions ι (i) : Z −→ Z, i = 0, 1 by : n ≥ 0, n − odd; Proof. Let n ≥ 1 be odd. Then If n ≥ 0 is even then Let i = 0 and n ≤ −1 or i = 1 and n < −1. Then In the remaining case i = 1 and n = −1 the formula follows directly from the definition.
To simplify notation we omitted the superscript ,,(i)" in the above lemma since the formula in this lemma is the same in both cases. In the sequel we will consistently omit this superscript.

Lemma 4.3. The members of S satisfy
Proof. Necessity is obvious.
Assume that k = l ± 1. Choosing the representation with the left lower entry equal to 1 we may write The statement now follows from Lemma 4.2.
Now we are in a position to show that the Cayley graphs Γ(C, S) are both isomorphic to our distant graph G. In order to do this let us define a map φ : We have shown that φ is a graph injection.
It is left to show that φ is an epimorphism. For a vertex y in some graph F denote a neighborhood of y in F by S y = {x ∈ V (F ) : x y}. With such a notation it is enough to show that S v ⊂ φ(C) for each v ∈ φ(C) since ± 1 0 ∈ φ(C) and our distant graph is connected.
Let φ(A) = v. We may enumerate neighborhoods S A = {A n : n ∈ Z} and S v = {v n : n ∈ Z} in such a way that A k A l and v k v l iff |k − l| = 1. This follows from the definitions or may be checked by inspection in Fig. 4. Since φ is an injection, φ(S A ) ⊂ S v and if φ(A k ) = v n k then v n k v n l iff |k − l| = 1. It follows that φ(S A ) = {v n k : k ∈ Z} = {v n : n ∈ Z} = S v ⊂ φ(C).
Let us denote: F k -the free group with k free generators; D l -the free product of l copies of Z 2 ; T m -the free product of m copies of Z 3 , k, l, m ∈ N 0 ∪ {ℵ 0 }. With such notations we obtained the following result. We are able to show even more (but it is not a subject of this paper), namely: Every group of the form F k * D l * T m < P SL(2, Z), where k ∈ {0, 2}, k + l + m = ℵ 0 , can serve as a Cayley representation of the distant graph of integers.
We conjecture that:

A subgroup of P SL(2, Z) is a group of Cayley representation of the distant graph of integers if and only if it is of the form
This will be a subject of a forthcoming paper.
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