Search

Let S be a regular semigroup, J(S) be the set of ideals of S and M be a subset of J(S). In this paper, we introduce an undirected Cayley graph of S, denoted by ΓS,M, with elements of J(S) as the vertex set, and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of M such that IK = J or JK = I. We study some basic properties of the graph ΓS,M such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of ΓS,M. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. Let Γ(R) denote the maximal graph associated to R, that is, Γ(R) is a graph with vertices as non-units of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we characterize the finite commutative rings such that their maximal graph are planar graphs, and we also study the case where they are outerplanar and ring graphs. The equivalence of outerplanar graphs and ring graphs for Γ(R) is established. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and I be a proper ideal of R. The annihilator graph of R with respect to I, which is denoted by AGI (R), is the undirected graph with vertex-set V (AGI (R)) = {x ∈ R \ I: xy ∈ I for some y ∉ I} and two distinct vertices x and y are adjacent if and only if AI (xy) ≠ = AI (x) ∪ AI (y), where AI (x) = {r ∈ R : rx ∈ I}. In this paper, we study some basic properties of AGI (R), and we characterise when AGI (R) is planar, outerplanar or a ring graph. Also, we study the graph AGI (Zn), where Zn is the ring of integers modulo n. [ABSTRACT FROM AUTHOR]