The endomorphisms monoids of Helm graph and its generalization

Let $G$ be a graph. Then $G$ is said to be End-regular if the set of all endomorphisms of $G$ forms a regular monoid. In this paper, we discuss the End-regularity of Helm graphs and our generalization. We also prove that the generalized Helm graph is End-orthodox if and only if it is End-regular. Moreover, we investigate the End-regularity of the join of two generalized Helm graphs