Suborbital graphs obtained by the modular congruence subgroup $\Gamma_0(L,M)$

TL;DR

This paper explores the action of suborbital graphs derived from the modular congruence subgroup (L,M), introducing new theorems and connecting these structures to various mathematical and physical disciplines.

Contribution

It develops new theorems on suborbital graphs associated with (L,M), expanding understanding of modular groups connected to two numbers and their applications.

Findings

New theorems on (L,M) suborbital graphs

Analysis of congruence relations beyond identity

Connections to algebraic geometry, number theory, and physics

Abstract

In the suborbital graphs studies, there has been a research gap in the sense that the Modular group is connected to two numbers. Thus, this paper attempts to contribute to the studies developed by Gauss, Bolyai, Lobachevsky and Riemann. However, this study mainly concentrates on the action of suborbital graphs obtained with the Modular congruence subgroup $\Gamma_0(L,M)$, making this study sui generis since it deals with the Modular group, connected to two numbers. In developing our graph action, we utilized the theories of non-Euclidean geometry. Investigating the congruence relation other than identity and universal relation, the number of congruence relation, transitive act on vertices and edges, edge condition for the congruence group $\Gamma_0(L,M)$, based on previously-obtained studies, we concluded with new theorems in this study. So, the results are obtained in this paper related to a different congruence modular subgroup provides various aspects of the same structure in mathematics and adapting it to such as algebraic geometry, number theory, differential geometry, topology and physics. Keywords: Modular group, Mobius transforms, suborbital graph