Structures and applications of graphs arising from congruences over moduli

For any positive integer $ \mathrm{n} $, let $ \mathrm{M_{p}} $ contain the prime numbers less than $ \mathrm{n} $. Assuming $ \mathrm{M_{p}} $ as the set of moduli, we draw a graph with the vertex set $ \{1, 2, 3, \cdots, \mathrm{n}\} $, and an edge will be built between the vertices $ \mathrm{p} $ and $ \mathrm{q} $ if and only if $ \mathrm{p}\equiv (\mathrm{q}\; mod\; \mathrm{m}) $ for some $ \mathrm{m}\in $ $ \mathrm{M_{p}}. $ We call this graph a prime congruence simple graph and label this graph as $ \mathrm{G}(\mathrm{n}, \mathrm{M}_{p}) $. The objective of this work is to characterize prime congruence simple graphs, and afterwards, by utilizing these graphs, solutions to the system of linear congruences are suggested and demonstrated by applying modular arithmetic. It is shown that this graph is always a connected graph. The generalized formulae for vertex degrees, size, chromatic number, domination number, clique number, and eccentricity of the prime congruence simple graphs are proposed and proved. Also, independence numbers as well as a covering number for the proposed graph through vertices and edges are evaluated. Lastly, as an application of prime congruence simple graphs, the solution of a system of linear congruences is discussed in terms of the degrees of the vertices.