Connectedness in Projection Graphs of Commutative Rings

Abstract

Background/Objectives: Connectedness plays an essential role in applications of graph theory. Connected graphs are used to represent social networks, transportation networks, and communication networks. In this study, various classes of commutative rings are explored to identify connected projection graphs. Methods: An investigation is carried out to exclude classes of rings whose projection graphs possess isolated vertices and connectedness is shown by establishing spanning subgraphs. Findings : Unipotent units and zero-divisors are found non-isolated, in which zero-divisors include idempotents and nilpotents. The element 2 is isolated if it is invertible. Necessary condition for the rings to have connected projection graphs is derived. Projection graphs of finite Boolean rings are connected and Hamiltonian. The local rings of integers modulo n with even characteristics are connected and contain spanning bistars. A criterion for certain class of nonlocal rings to have connected projection graphs is described. Novelty: Study on projection graphs in the perception of unipotent units is carried out, which is not done earlier in any other algebraic graph.