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

Let G be a group with the neutral element e. If R = circle plus(g is an element of G) R-g is a G-graded ring with unity 1, then it is well-known that 1 is an element of R-e. In general, let Delta be a cancellative magma and R = circle plus(delta is an element of Delta) R-delta Delta-graded ring with unity 1. The set boolean OR(delta is an element of Delta) R-delta of all of its homogeneous elements is denoted by H-R, and by I (Delta) we denote the set of all idempotent elements of Delta. Then for each epsilon is an element of I (Delta), the subring R-epsilon is with unity 1(epsilon), and, moreover, for every x is an element of H-R there exist xi, eta is an element of I (Delta) such that 1(xi) x = x = x1(eta). Let R be a Delta-graded ring with these two properties (for instance, a group-graded ring). By G(H-R) we denote the undirected power graph of a multiplicative subsemigroup HR of R, and by G degrees (H-R) the graph obtained from G(H-R) by removing all nonzero vertices 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 G degrees(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 as it implies that the graph G degrees (H-R) is connected if and only if its set of vertices forms a nil semigroup.