Kind of bipartite graph associated on triple elements of subgroup of group

Abstract

Assume $\mathbb{G}$ is a non commutative group, $\mathbb{H}$ is a subgroup of $\mathbb{G}$, in this paper we define a kind of bipartite graph which denoted by ${\ \ \Delta }_{ \mathbb{H},\mathrm{\ }\mathbb{G}} $ and define as a vertex set $V =MUN\ $ were $\ M=\{(r ,r ,r )\in \mathbb{H}\times \mathbb{H}\times \mathbb{H}\}\ /\{(r ,r ,r )\in {\mathbb{H}}^3:[t,3r ]=1,\ \forall t\in \mathbb{G}\}\ \&\ N=\ \mathbb{G}-\{t\in \mathbb{G}:[t,3r ]\ =1,\ \forall r \in \mathbb{H}\}$. Two vertices $t$ and $(r ,r ,r )$ are adjacent iff $\left[t,r ,r ,r \right]\neq 1 $ this graph has no isolated vertex ,also we introduce the relative 3-Engle degree of $\mathbb{H}$ in $\zeta$ and discuss the relation between it and the graph ${\ \ \Delta }_{\mathbb{H}\mathrm{,\ }\mathbb{G}}$.