On the Connected Power Graphs of Semigroups of Homogeneous Elements of Graded Rings

Abstract

In this paper, by the power graph ${\mathcal {G}}(S)$ of a semigroup S we mean an undirected graph whose vertices are elements of S and where two vertices are adjacent if and only if they are distinct and one of them is a power of the other. Let $R=\bigoplus _{s\in S}R_s$ be a ring graded by a groupoid S. Inspired by the problems raised in Abawajy et al. (Electron J Graph Theory Appl 1(2):125–147, 2013) we investigate the question of connectedness of the power graph of the multiplicative semigroup $H_R=\bigcup _{s\in S}R_s$ of homogeneous elements of R. We establish that ${\mathcal {G}}(H_R)$ is connected if and only if all of the homogeneous elements of R are nilpotent. If ${\mathcal {G}}(H_R)$ is connected, then the power graphs ${\mathcal {G}}(R_e)$ of the multiplicative semigroups $R_e,$ where e runs through the set of all idempotent elements of S,  are also connected. The converse, however, does not hold in general, but we prove that it does hold under some additional assumptions. If R has no nontrivial homogeneous right or left zero divisors, then $H_R^*=H_R{\setminus }\{0\}$ is a semigroup under the multiplication of R,  and S is a semigroup. If, moreover, R is with unity and S is cancellative, we prove that ${\mathcal {G}}(H_R^*)$ is connected if and only if S is a monoid with unity e,  and the power graphs ${\mathcal {G}}(R_e{\setminus }\{0\})$ and ${\mathcal {G}}(S)$ are connected.