Sharif Digital Repository

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...