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

Let S be a groupoid (magma) with zero 0, and let R=⊕s∈SRs be a contracted S-graded ring, that is, an S-graded ring with R0=0. By G(HR) we denote the undirected power graph of a multiplicative subsemigroup HR=∪s∈SRs of R, and by G*(HR)a graph obtained from G(HR) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G*(Re) denotes a subgraph of G*(HR), induced by Re*. In this paper 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). Namely, let S be torsion-free, that is, sn=tn implies s = t for all s, t∈S, and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u∈S,su=tu≠0 implies s=t, and us=ut≠0 implies s=t. Also, let R be semisimple Artinian. We prove that if G*(Re) is connected for every nonzero ring component Re of R, then the connected components of G*(HR) are precisely the graphs G*(Re).