Abstract
This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.
Highlights
E use of graphs to represent group actions has a venerable history
Mathematicians like Coxeter [9], Burnside [10], Stothers [11], Everitt [12], Conder [13], Whitehead [14], and others provided pioneering works on graphical representations of groups. e action of a modular group on certain objects can be represented by a certain type of graphs, called coset graphs. ese were introduced by Higman in 1978
In 1983, Mushtaq [15] laid their foundation. ese graphs consist of triangles connected to each other. e edges of triangles are permuted anticlockwise to represent by g
Summary
Hgh− 1 g− 1− 1, h turns around the direction of the triangles like reflection. us, we do not introduce h-edges in coset graphs, so that they remain simple. Let a homomorphic copy c of C be formed by contracting its vertices u1 and u2. 3. Formation of Homomorphic Copies through Contraction of Vertices e coset graphs are made up of closed paths. Erefore, a question arises: how many distinct homomorphic copies can be created by contracting all pairs in a closed path? If c is obtained by contracting vertices u1 and u2 of any closed path C, the mirror image c∗ of c can be created by Figure 4: A homomorphic copy of (4, 3). We contract a pair of vertices u1, u2 of ψ such that a homomorphic copy c is obtained. It can 1)i1 f and be seen that g− 1(fg− 1)m1− i1− 1 are the possible paths between s3m1 and s3i1+1. If vertex s3m1 is contracted with vertices t3i1+1, Fthuernthmer2mdoirset,inthcterheoamreom3(omrp22h+ic3cmop2)iepsaβiri1s of of ψ are obtained. vertices for these homomorphic copies
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