A graph associated to a polygroup with respect to an automorphism

In this paper, we introduce and study, $zeta^alpha(P)$, the $alpha$-center of a polygroup $(P, cdot )$ with respect to an automorphism $alpha$. Then we associate to $P$ a graph $Gamma^alpha_{P}$, whose vertices are elements of $P setminus zeta^alpha(P)$ and $x$ connected to $y$ by an edge in case $x cdot y cdot omega neq y cdot x^alpha cdot omega $ or $y cdot x cdot omega neq x cdot y^alpha cdot omega$, where $omega $ is the heart of $P$. We obtain some basic properties of this graph. In particular, we prove that if $zeta^alpha(P) neq P$, then $dim(Gamma^alpha _{P})=2$. Moreover, we define a weak $alpha$-commutative polygroup to state that if $Gamma^alpha_{H} cong Gamma^beta_{K}$ and $H$ is a weak $alpha$-commutative, then $ K$ is a weak $beta $-commutative. Also, we show that if $H$ and $K$ are two polygroups such that $Gamma^alpha_{H} cong Gamma^beta_{K}$, then for some automorphisms $eta$ and $lambda$, $Gamma^eta_{H times A} cong Gamma^lambda_{K times B}$, where $A$ and $B$ are two weak commutative polygroups.