On the power graphs of semigroups of homogeneous elements of graded rings

Let G be a group with the neutral element e. If \(R=\bigoplus _{g\in G}R_g\) $R={⨁}_{g\in G}{R}_g$ is a G-graded ring with unity 1,  then it is well-known that \(1\in R_e.\) $1\in R_e.$ In general, let \(\Delta \) $\mathrm{\Delta }$ be a cancellative magma and \(R=\bigoplus _{\delta \in \Delta }R_\delta \) $R={⨁}_{\delta \in \mathrm{\Delta }}{R}_{\delta }$ a \(\Delta \) $\mathrm{\Delta }$ -graded ring with unity 1. The set \(\bigcup _{\delta \in \Delta }R_\delta \) ${\bigcup }_{\delta \in \mathrm{\Delta }}{R}_{\delta }$ of all of its homogeneous elements is denoted by \(H_R,\) $H_R,$ and by \(I(\Delta )\) $I(\mathrm{\Delta })$ we denote the set of all idempotent elements of \(\Delta .\) $\mathrm{\Delta }.$ Then for each \(\varepsilon \in I(\Delta ),\) $\epsilon \in I(\mathrm{\Delta }),$ the subring \(R_\varepsilon \) $R_{\epsilon }$ is with unity \(1_\varepsilon ,\) $1_{\epsilon },$ and, moreover, for every \(x\in H_R\) $x\in H_R$ there exist \(\xi ,\) $\xi ,$ \(\eta \in I(\Delta )\) $\eta \in I(\mathrm{\Delta })$ such that \(1_\xi x=x=x1_\eta .\) $1_{\xi }x=x=x{1}_{\eta }.$ Let R be a \(\Delta \) $\mathrm{\Delta }$ -graded ring with these two properties (for instance, a group-graded ring). By \(\mathcal {G}(H_R)\) $\mathcal{G}(H_R)$ we denote the undirected power graph of a multiplicative subsemigroup \(H_R\) $H_R$ of R,  and by \(\mathcal {G}^\circ (H_R)\) ${\mathcal{G}}^{\circ }(H_R)$ the graph obtained from \(\mathcal {G}(H_R)\) $\mathcal{G}(H_R)$ by removing all nonzero vertices \(1_\varepsilon \) $1_{\epsilon }$ and their incident edges. We address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power graphs: a survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)], by characterizing the connectedness of the graph \(\mathcal {G}^\circ (H_R)\) ${\mathcal{G}}^{\circ }(H_R)$ in terms of the nonzero subring components of R. This characterization turns out to be closely related to the nil radical of \(H_R\) $H_R$ as it implies that the graph \(\mathcal {G}^\circ (H_R)\) ${\mathcal{G}}^{\circ }(H_R)$ is connected if and only if its set of vertices forms a nil semigroup.