Some operations on congruence labeled graphs

Abstract

A graph \(G(V,E)\) is identified as congruence graph, if its vertices and edges are labeled by distinct natural numbers, which induces \(f(x_i) \equiv f(x_j ) \pmod g(y)\),  where each edge \(x_{i}x_{j}\in E\) with \(x_{i},x_{j}\in V\). In this paper, some operations were performed on congruence graphs such as union of congruence path graphs & \(C_{m}\cup P_{n}\) are identified as congruence graph. Furthermore, graphs procured by duplicating any arbitrary vertex, edge in a cycle \(C_{n}\) and cartesian product of two path are confirmed as congruence graph.