THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-I

Abstract

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by T Γ (R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of T Γ (R) where R is a commutative Artin ring. The intersection graph of gamma sets in T Γ (R) is denoted by I TΓ (R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph I TΓ (ℤ n ) of gamma sets in T Γ (ℤ n ). In this paper, we study about I TΓ (R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of I TΓ (R) and ring-theoretic properties of R. At the first instance, we prove that diam (I TΓ (R)) ≤ 2 and gr (I TΓ (R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of I TΓ (R) are given. Further, we discuss about the vertex-transitive property of I TΓ (R). At last, we obtain all commutative Artin rings R for which I TΓ (R) is either planar or toroidal or genus two.