Suborbital graphs for the group Gamma

Abstract

In this paper, we investigate suborbital graphs formed by the action of Γ 2 which is the group generated by the second powers of the elements of the modular group Γ onQ. Firstly, conditions for being an edge, self-paired and paired graphs are provided, then we give necessary and sufficient conditions for the suborbital graphs to contain a circuit and to be a forest. Finally, we examine the connectivity of the subgraph Fu,N and show that it is connected if and only if N ≤ 2.

Suborbital graphs of a power subgroup of the modular group

“…Actually, the suborbital graphs of the group Γ 2 were studied in [7] for the relation Γ 2 ∞ Γ 2 0 (n) Γ 2 with n ∈ N. In here, taking Γ (2) instead of Γ 2 0 (n), we investigate some combinatorial properties of the newly constructed subgraphs of Γ (2) different from [7]. We can summarize the cause of this choice as follows.…”

“…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.…”

Solutions to Pell's Equation via Suborbital Graphs

“…Sims introduced the concept of suborbital graphs for finite permutation groups in [16]. Following that, many studies focused on suborbital graphs for the modular group and similar objects, it can be seen [14,15,7,8,3]. Also we can mentioned more studies such as Kader e.t.…”