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 if ζα(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.