The non-zero divisor graph of ring of integers modulo six and the Hamiltonian quaternion over integers modulo two

Abstract

The study of graph theory was introduced and widely researched since many practical problems can be represented by graphs. A non-zero divisor graph is a graph in which its set of vertices is the non-zero elements of the ring and the vertices x and y are adjacent if and only if xy ≠ 0. In this study, we introduced the non-zero divisor graphs of some finite commutative rings in specific the ring of in tegers modulo 6, ℤ6 and ring of Hamiltonian quaternion, ℍ(ℤ2). First, the non-zero divisors of the commutative rings are found. Then, the non-zero divisor graphs are constructed. Finally, some properties of the graph, including the chromatic number, clique number, girth and the diameter are obtained.