Let (G, *) be a finite group and S = {x ∈ G|x = x −1 } be a subset of G containing its non-self invertible elements. The inverse graph of G denoted by (G) is a graph whose set of vertices coincides with G such that two distinct vertices x and y are adjacent if either x * y ∈ S or y * x ∈ S. In this paper, we study the energy of the dihedral and symmetric groups, we show that if G is a finite non-abelian group with exactly two non-self invertible elements, then the associated inverse graph (G) is never a complete bipartite graph. Furthermore, we establish the isomorphism between the inverse graphs of a subgroup D p of the dihedral group D n of order 2p and subgroup S k of the symmetric groups S n of order k! such that 2p = n! (p, n, k ≥ 3 and p, n, k ∈ Z +).

Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of of this Γ1 non-deranged permutation group p  (p ≥ 5 and p a prime). Also we reveal some interesting properties and results of the character () i χ ω of p  where i p ω ∈  .

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.