Unit Regular Graphs over Finite Rings

This paper introduces the concept of a unit regular graph over any given ring. The unit regular graph is a simple undirected graph, where its vertices correspond to elements of the ring. Two vertices are connected if their sum results in a unit regular element within the ring. The research explores various aspects of this graph, such as completeness, vertex degrees, Eulerian and Hamiltonian properties, girth, matching number, clique number, and independence number. Furthermore, the study delves into specific properties of these graphs for certain ring types.