A note on co-maximal graphs of commutative rings

Abstract

Let R be a commutative ring with unity. The co-maximal graph Γ ( R ) is the graph with vertex set R and two vertices a and b are adjacent if R a + R b = R . In this paper, we characterize rings for which the co-maximal graph Γ ( R ) is a line graph of some graph G and determine rings for which Γ ( R ) is Eulerian or Hamiltonian. We also find the diameter and girth of Γ ( R ) . Keywords: Finite commutative ring, Maximal ideal, Co-maximal graph, Line graph