Let S be a homogeneous (graded) completely simple semigroup, graded by a cancellative semigroup \(\Delta ,\) that is, S is a completely simple semigroup with a family \(\{S_\delta \}_{\delta \in \Delta }\) of mutually disjoint nonempty subsets of S, called components, such that \(S=\bigcup _{\delta \in \Delta }S_\delta \) and \(S_\delta S_\gamma \subseteq S_{\delta \gamma }\) for all \(\delta ,\) \(\gamma \in \Delta .\) We prove that \(\Delta \) is a group, and that S takes the form of the Rees matrix semigroup over a \(\Delta \) -graded group G, and with sandwich matrix whose entries are from \(G_\varepsilon ,\) where \(\varepsilon \) is the identity element of \(\Delta .\) Let \(\mathcal {G}(S)\) be the power graph of S, that is, an undirected graph with S as the set of vertices, and where two distinct vertices are adjacent if and only if one is a power of the other. Then the vertex set of \(\mathcal {G}(S)\) is a disjoint union of vertex sets of its subgraphs induced by \(S_\delta ,\) called the homogeneous components of \(\mathcal {G}(S),\) for all \(\delta \in \Delta .\) Let S and T be finite homogeneous completely simple semigroups whose \(\mathcal {H}\) -classes are Abelian, let \(\varphi :\mathcal {G}(S)\rightarrow \mathcal {G}(T)\) be a graph isomorphism which maps each homogeneous component of \(\mathcal {G}(S)\) onto a homogeneous component of \(\mathcal {G}(T),\) and let there exist an \(\mathcal {R}\) -class and an \(\mathcal {L}\) -class of S which are mapped by \(\varphi \) onto an \(\mathcal {R}\) -class and an \(\mathcal {L}\) -class of T, respectively. Then we prove that S and T are graded by the same group, and that there exists a semigroup isomorphism \(f:S\rightarrow T\) which preserves their gradings. We also investigate the relationship between the connectedness of the power graph and of the proper power graph of a completely regular semigroup, which is moreover homogeneous, and the connectedness of the subgraphs induced by the group components of its \(\mathcal {H}\) -classes.
