The intersection graph of gamma sets in the total graph of a commutative ring-I

摘要

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. Algebra 320 (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 Theory 32 (2012) 339-354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph I_(TΓ) (_n) of gamma sets in T _Γ(_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.