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Co-maximal Graph of Algebraic Structures

Miraftab, Babak | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 44608 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saieed
  7. Abstract:
  8. In this thesis, we study some connections between the graph-theoretic and algebraic properties of co-maximal graph of algebraic structures. We follow two purposes. First, what properties of algebraic structures can be found from co-maximal graph of algebraic structures. Second, what geometric or graph theoretical properties of co-maximal graph of algebraic structures can be found from specefic algebraic structures. Let G be a group and I(G)∗be the set of all non-trivial sub-groups of G. The co-maximal graph of subgroups of G, denoted byΓ(G), is a graph with the vertex set I(G)∗and two distinct vertices H and K are adjacent if and only if HK=G. We char-acterize all groups whose co-maximal graph of subgroups are connected. Among other results, the diameter of Γ(G) for all groups G has been computed. Also, we have characterized Abelian groups for which the diameter of co-maximal graphs of subgroups is two. We completely answer this question for finite groups. When Γ(G) is a tree, for a given group? Also, we have characterized all finitely generated Abelian groups for which the co-maximal graph of subgroups is a forest. Finally,we find the clique number and chromatic number of co-maximal graph of subgroups of finitely generated Abelian groups. Let R be a commutative ring with unity. The vertices are proper and non-trivial ideals of R which are not in the Jacobson radical and two distinct vertices I and J are adjacent if and only if I+J=R. The co-maximal graph of ideals of ring, denoted by Γ(R). We characterize all rings whose co-maximal graph of ideals are connected. We discuss about some properties of co-maximal graph of ideals of ring. For instance, we find the diameter and the girth of co-maximal graph of ideals. Also, we have characterized all rings for which the co-maximal graph of ideals is a complete bipartite graph or an n-partite graph. The definition of co-maximal graph of submodules of a module is like co-maximal graph of subgroups. Let M be a unitary R-module and I(M)∗ is the set of all non-trivial submodules of M. The co-maximal graph of submodules of M, denoted byΓ(M), is a graph with the vertex set I(M)∗ and two distinct vertices N and L are adjacent if and only if N+L=M. We characterize all modules whose co-maximal graph of submodules are connected. Among other results, the diameter and girth of co-maximal graph of submodules have been computed. We discuss about the relation of co-maximal graph of submodules of module with co-maximal graph of submodules of endomorphism ring of module.For example, we discuss about the connectivity and nullity of those graphs. Also, we have charac-terized all trees for which the co-maximal graphs of subgroups is tree. In the end, we show that if the veretex of the co-maximal graph of submodules is a pendant vertex, then the neighborhood of that submodule is a fully invarient submodule
  9. Keywords:
  10. Co-Maximal Graph of Subgroups of a Group ; Co-Maximal Graph of Ideals of Ring ; Co-Maximal Graph of Submodules of Module ; Nilpotent Group ; Super Solvable Group ; Projective Module ; Clique Number ; Chromatic Number ; Complete Graph

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