Intersection Graph

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Yaghoobian, Maysam | 2011

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 42430 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
Let R be a ring with unity, M be a unitary left R-module and I(M)* be the set of all non-trivial submodules of M. The intersection graph of submodules of M, denoted by G(M), is a graph with the vertex set I(M)* and two distinct vertices N and K are adjacent if and only if N\K ̸= 0. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph. First we start our study with a simple case, when M = R. We characterize all rings whose intersection graph of ideals are not connected. Also we determine all rings whose clique number of the intersection graph of ideals are finite
Keywords:
Chromatic Number ; Clique Number ; Intersection Graph