Cohen–Macaulayness of a Class of Graphs Due to Grimaldi<br/>

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Cohen–Macaulayness of a Class of Graphs Due to Grimaldi

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 55042 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Pournaki, Mohammad Reza
Abstract:
Let K be a field and S=K[x0,…,xn-1] be the polynomial ring in n variables over the field K. Let G be a finite undirected graph without loops or multiple edges with the vertex set V(G)={0,…,n-1} and the edge set E(G). One can associate a squarefree quadratic monomial ideal I(G)=of S to the graph G. The ideal I (G) is called the edge ideal of G in S. It is an algebraic object whose invariants can be related to the properties of G and vice versa. The graph G is called Cohen–Macaulay over K (Gorenstein over K) if the ring S/I (G) is Cohen–Macaulay (Gorenstein). Let n ≥ 2 be an integer. The Grimaldi graph represented by 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,1)=1.In this thesis, structural properties, wellcoveredness, Cohen–Macaulayness, vertexdecomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs
Keywords:
Cohen-Macualay Modules ; Well-Covered Graph ; F -Vector ; Grimaldi Graph ; Corenstein Graph ; Vertex Decomposability Graph