On the Intersection Graphs Associeted to Posets

Let (P, ≤) be a poset with the least element 0. The intersection graph of ideals of P, denoted by G(P), is a graph whose vertices are all nontrivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ {0}. In this paper, we study the planarity and outerplanarity of the intersection graph G(P). Also, we determine all posets with split intersection graphs

On the associated graphs to a commutative ring

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by Gamma(S)(R) with all elements of R as vertices, and two distinct vertices x, y is an element of R are adjacent if and only if x + y is an element of S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of Gamma(S)(R). Moreover, we will improve and generalize some results for the total and the unit graphs