a graph associated to a polygroup with respect to an automorphism

In this paper, we introduce and study, ζα(p ), the α-center of a polygroup (p, ·) with respect to an automor-phism α. then we associate to p a graph γpα , whose vertices are elements of p ζα(p ) and x connected to y by an edge in case x · y · ω = y · xα · ω or y · x · ω = x · yα · ω, where ω is the heart of p . we obtain some basic prop-erties of this graph. in particular, we prove that ifζα(p ) = p , then dim(γpα ) = 2. moreover, we define a weak α-commutative polygroup to state that if γα∼=γβ and h is a weak α-commutative, then k is a weak β-commutative. also, we show that if h and k are two polygroups such that γα ∼= γ , then for some automor-phisms η and λ, γηh×a∼=γλk×b, where a and b are two weak commutative polygroups.