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Some Graphs Associated with Groups

Rahimi Rad, Saman | 2016

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48633 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. In this thesis we ascribe a one graph to groups and investigate the relations between groups structure and graph. Let G be a group. The prime index graph of G, denoted by π(G), is the graph whose vertex set is the set of all subgroups of G and two distinct comparable vertices H and K are adjacent if and only if the index of H in K or the index of K in H is prime.Also, The intersection graph of G denoted by Γ(G), is the graph whose vertex set is the set of all non-trivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ⋂ K≠ {e}, where e denoted the identity element of G. We denote the complement of Γ(G) by Λ(G). In this thesis we consider the following results: We show that the diameter of every connected component of Λ(G) does not exceed 3.We characterize all abelian groups whose complememt of intersection graphs are connected.Moreover, we show that if Λ(G) is triangle-free, then Λ(G) is bipartite graph. Also, we prove that for abelian group G, Λ(G) is a triangle-free graph, if and only if G is isomorphic to a subgroup of a subgroup of Q × Zp∞ or Zp∞ × Zq∞, for some distinct prime numbers p and q. Also, we classify all finite groups of odd orders whose intersection graphs are triangle-free.We determine the domination number of Γ(G), whenever G is an abelian group or a finite nilpotent group that is not a p−group, where p is a prime. We show that the domination number of the intersection graph of a nilpotent groups is finite. Also we show that For every group G, Π(G) is bipartite and the girth of Π(G) is contained in the set{4,∞}. For any finite abelian group G, Π(G) is a regular graph if and only if Π(G) is a hypercube graph. We also show that for every finite solvable group G, Π(G) is a connected graph. Also we prove that if G is a finite solvable group, then Π(G) is connected. we prove that if Π(G) is a connected graph and N is a normal subgroup of G, then both graphs Π(N) and Π(G/N) are connected
  9. Keywords:
  10. GROUPS ; Bipartite Graph ; Intersection Graph ; Girth Graph ; Domination Number ; Prime Index Group

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