Number of Distinct Homomorphic Images in Coset Diagrams

The representation of the action of PGL2,Z on Ft∪∞ 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 bypqΔ1pq−1Δ2pqΔ3…pq−1Δm∈PSL2,Z, then this circuit is titled to be a length-m circuit, denoted byΔ1,Δ2,Δ3,…,Δm. In this manuscript, we consider a circuit Δ of length 6 as Δ1,Δ2,Δ3,Δ4,Δ5,Δ6 with vertical axis of symmetry, that is, Δ2=Δ6,Δ3=Δ5. Let Γ1 and Γ2 be the homomorphic images of Δ acquired by contracting the vertices a,u and b,v, respectively, then it is not necessary that Γ1 and Γ2 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 Δ1>Δ2>Δ3>Δ4. 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.