Circuit lengths of graphs for the Picard group

Abstract

In this paper, we examine some properties of suborbital graphs for the Picard group. We obtain edge and circuit conditions, then propose a conjecture for the graph to be forest. This paper is an extension of some results in (Jones et al. in The Modular Group and Generalized Farey Graphs, pp. 316-338, 1991).

Suborbital graphs for the group Gamma

“…The graph Gu,N is a forest if and only if u 2 ± u + 1 ≡ 0 (mod N ). Case F2,3: As above, let's show that no vertices of F2,3 between 3 2 and 2 are adjacent to vertices outside this interval. Suppose that 1 ≤ x 3y < 3 2 < a 3b < 2 and x 3y < −→ a 3b ∈ F2,3.…”

“…Akbas proved in [2] that this conjecture is true. By similar arguments, we concern with suborbital graphs of Picard group P, which is the subgroup of PSL(2, C) with entries coming from Z[i] in [3]. Since Z[i] is a unique factorization domain with finitely many units, our expectation was to find similar formulas.…”

Connectedness of suborbital graphs for a special subgroup of the modular group

“…A subgroup of Γ is called a congruence group provided it contains the principal congruence group Γ(n).Congruence groups have been of great interest in many fields of mathematics, number theory, group theory, etc. Jones, Singerman and Wicks [1] used the notion of the imprimitive action [3], [4], [9] for a Γ− invariant equivalence relation induced on Q ∪ {∞} by the congruence subgroup Γ 0 (n) = a b c d ∈ Γ : c ≡ 0 (mod n) to obtain some suborbital graphs and examined their connectedness and forest properties.Some applications of this method can be found in the papers, [1], [2].Particularly in [2], [12], [13] and [14] authors give some results about a connection between the periods of elliptic elements of chosen permutation group with the circuits in suborbital graphs of it. In this article we introduce a different invariant equivalence relation by using the congruence subgroup…”

Suborbital graphs of a power subgroup of the modular group

“…These ideas were first introduced by Sims [15] and are also described in a paper by Newmann [10] and in books Tsuzuku [17], Biggs and White [4], the emphasis being on applications to finite groups. The reader is also refereed to [2][3][6] [7] for some relevant previous work on suborbital graphs. In the opposite direction, we shall prove the theorem for minus sign.…”