Structural Properties of Graphs Defined by Bilinear Congruence on the Ring of Integers Modulo n

We investigate the structural properties of a graph defined on the ring . The Adjacency between two different vertices and is determined by the bilinear congruence . We analyze three fundamental cases, and for distinct odd primes . We describe the graph's breakdown into unit and non-unit vertex subsets. The unit subgraph forms disjoint cliques, with sizes depending on Euler's totient function. In contrast, the zero-divisor subgraph shows more complex behaviour governed by annihilation ideals. We establish general properties, including degree formulas, determination of maximum clique sizes in each component, determining the diameter, computing the girth, locating the graph centers, and finding the measures of vertex and edge connectivity. Additionally, we characterize independent sets and prove the existence of Hamiltonian cycles and supereulerian properties under certain connectivity conditions. Our results show how the prime factorization of influences these properties.