In recent years, graph theory has become much more popular due to its wide range of applications in almost every field of interest. Especially, it plays an integral role in cryptography and coding theory to develop new security systems. In this article, we introduce and investigate novel discrete graphs arising from symmetric groups and incorporate these graphs to communicate confidential messages via an interesting encryption. These newly defined discrete graphs are termed as Order Incongruent Simple Graphs (OISG). The vertices of these graphs are the permutations of the symmetric groups, and edges are defined via modular arithmetic. Many fascinating structures and characterizations of these graphs are proposed and proved using congruences from number theory.