Commutative Rings and Corresponding V-Graphs

Algebraic graph theory, which applies algebraic techniques to graph problems, is a pivotal area of study. Many ring algebraic properties have representations in graph theory. In this paper, we introduce an innovative type of graph related to rings that we call the V-graph. Let T be a ring with identity. The V-graph of T, denoted by V(T), consists of a set of vertices equal to all non-zero elements of T. Two different vertices u and v are adjacent if and only if uv is a regular element in T. We present several examples to demonstrate how this form of graph differs from the known ring-based graphs, such as zero-divisor graphs, unit graphs, and quasi-regular graphs. We calculate the diameter and independent number, as well as the domination number for the V-graph of the ring of integers Zn.