Structural Properties of a Class of Cayley Graphs and their Complements: Well-coveredness and Cohen–Macaulayness

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Khezerlou, Parian | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54741 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Pournaki, Mohammad Reza
Abstract:
Let be a field and R= [x0, . . . , xn−1] be the polynomial ring in n variables over the field . Let G be a finite simple graph with the vertex set V(G) ={0, . . . , n − 1} and the edge set E(G). One can associate a square-free quadratic monomial ideal I(G) = (xixj | {i, j} ∈ E(G)) of R to the graph G. The ideal I(G) is called the edge ideal of G in R. The graph G is called Cohen–Macaulay (resp. Buchsbaum, Gorenstein) over if the ring R/I(G) is Cohen–Macaulay (resp. Buchsbaum, Gorenstein).Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this thesis, we investigate structural properties of Cayley graphs generated by the Sn’s and their complements. In particular, we classify all the graphs of these classes that are well-covered, Cohen–Macaulay, Buchsbaum or Gorenstein
Keywords:
Cayley Graph ; Cohen-Macualay Modules ; Gorenstein Rings ; F -Vector ; Buchsbaum Graph ; Well-Covered Graph