Number of Distinct Homomorphic Images in Coset Diagrams
The representation of the action of PGL 2 , Z on F t ∪ ∞ in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by p q Δ 1 p q − 1 Δ 2 p q Δ 3 … p q − 1 Δ m ∈ PSL 2 , Z , then this circuit is titled to be a length- m circuit, denoted by Δ 1 , Δ 2 , Δ 3 , … , Δ m . In this manuscript, we consider a circuit Δ of length 6 as Δ 1 , Δ 2 , Δ 3 , Δ 4 , Δ 5 , Δ 6 with vertical axis of symmetry, that is, Δ 2 = Δ 6 , Δ 3 = Δ 5 . Let Γ 1 and Γ 2 be the homomorphic images of Δ acquired by contracting the vertices a , u and b , v , respectively, then it is not necessary that Γ 1 and Γ 2 are different. In this study, we will find the total number of distinct homomorphic images of Δ by contracting its all pairs of vertices with the condition Δ 1 > Δ 2 > Δ 3 > Δ 4 . The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.