Subgroup Graphs of Finite Groups

Abstract

Let G be a fnite group with the set of subgroups of G denoted by S(G), then the subgroup graphs of G denoted by T(G) is a graph which set of vertices is S(G) such that two vertices H, K in S(G) (H not equal to K)are adjacent if either H is a subgroup of K or K is a subgroup of H. In this paper, we introduce the Subgroup graphs T associated with G. We investigate some algebraic properties and combinatorial structures of Subgroup graph T(G) and obtain that the subgroup graph T(G) of G is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups. Let be a finite group with the set of subgroups of denoted by , then the subgroup graphs of denoted by is a graph which set of vertices is such that two vertices , are adjacent if either is a subgroup of or is a subgroup of . In this paper, we introduce the Subgroup graphs associated with . We investigate some algebraic properties and combinatorial structures of Subgroup graph and obtain that the subgroup graph of is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups.