Leírás
The Cohen–Macaulay rings have a significant importance and have been studied by many authors in Commutative Algebra and Algebraic Geometry. One of the fundamental questions in the area is to classify Cohen–Macaulay quotient rings. Although Cohen–Macaulay rings keep appearing in Commutative Algebra, they are not easily recognizable. To deal with this issue, researchers study a combinatorial object associated with it. The combinatorial objects are called simplicial complexes. The results on this subject can help us to determine whether a given commutative ring is Cohen–Macaulay. Quite recently different authors focus their attention to the study of Cohen–Macaulayness of particular graphs.
Let n ≥ 2 be an integer. The Grimaldi graph G(n) is obtained by letting all the elements of {0,...,n-1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd(x + y,n) = 1. In this talk, we discuss the Cohen–Macaulayness of G(n) and its complement, namely G′(n).