Bounds for the Genus of Generalized Total Graph of a Commutative Ring.

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if x + y ∈ S (I) = { a ∈ R : r a ∈ I  for some  r ∈ R \ I }. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs. [ABSTRACT FROM AUTHOR]