Graphs Associated with Algebraic Structure

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Alizadeh, Faezeh | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 44304 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
One of the interesting and active area in the last decade is using graph theoretical tools to study the algebraic structures. In this thesis, first we study the intersection graphs of non-trivial submodules of a module, their clique number and their chromatic number. Next, we study the power graph of a group and observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs should be isomorphic. It also is shown that the only finite
group whose automorphism group is the same as that of its power graph is the Klein group of order 4. We study the cozero-divisor graph of R denoted by ′(R) and we show that if ′(R) is a forest, then ′(R) is a union of isolated vertices or a star graph. Also, we prove that if ′(R) is a forest with at least one edge, then R _=Z2_F, where F is a field. Finally, we study the unit graph of R, and it is shown that if R is a ring (not necessary commutative), then its unit graph is a complete r-partite graph if and only if (R;m) is a local ring and r = jR=mj = 2n, for some n 2 N
Keywords:
Clique Number ; Chromatic Number ; Local Ring ; Forest ; Complete Graph