A large class of graphs with a small subclass of Cohen–Macaulay members. Issue 12 (2nd December 2022)

• Record Type: Journal Article
• Authors: Ashitha, T.; Asir, T.; Pournaki, M. R.
• Is Part Of: Communications in algebra. Volume 50:Issue 12(2022)
• Page Start: 5080
• Page End: 5095
• Publication Date: 2022-12-02
• Subjects: Cohen–Macaulay graph -- finite commutative ring -- Gorenstein graph -- unit graph -- well-covered graph; Primary: 05C75 -- 13H10 -- Secondary: 05E40 -- 05E45; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2022.2081333
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R . In this paper, we determine when these graphs are well-covered, and then, by applying this result, we characterize the unit graphs whose edge rings are Cohen–Macaulay (Gorenstein). This characterization gives us a large class of non-Cohen–Macaulay graphs.