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 Γ ( 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 Γ ( 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 2 p and subgroup S k of the symmetric groups S n of order k ! such that 2 p = n ! ( p , n , k ≥ 3 and p , n , k ∈ ℤ + ) . $2p = n!~(p,n,k \geq 3~\text {and}~p,n,k \in \mathbb {Z}^{+}).$

中文翻译：

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二面体和对称群的逆图的能量

令 ( G ,∗) 为有限群且 S ={ x ∈ G | x ≠ x −1 } 是 G 的子集，包含其非自可逆元素。Γ ( G ) 表示的 G 的逆图是一个图，其顶点集与 G 重合，使得两个不同的顶点 x 和 y 相邻，如果 x ∗ y ∈ S 或 y ∗ x ∈ S 。在本文中，我们研究了二面体群和对称群的能量，我们证明如果 G 是一个具有恰好两个非自可逆元素的有限非阿贝尔群，那么关联的逆图 Γ ( G ) 永远不会是一个完整的二部图形。此外，我们建立了 2 p 阶二面体群 D n 的子群 D p 和 k 阶对称群 S n 的子群 S k 的逆图之间的同构！使得 2 p = n ！( p , n , k ≥ 3 并且 p , n , k ∈ ℤ + ) 。$2p = n!~(p,n,k \geq 3~\text {and}~p,n,k \in \mathbb {Z}^{+}).$