For a real number \(q>0\) , the q-th outdegree power of a digraph D is \(\partial _{q}^{+}(D)=\sum _{v\in V(G)}(d_{D}^{+}(v))^{q}\) . For a graph G, \({\mathcal {D}}(G)\) is the set of all orientations of G. We focus on a fundamental problem on deciding \(\min \{\partial _{q}^{+}(D): D\in {\mathcal {D}}(G)\}\) and \(\max \{\partial _{q}^{+}(D): D\in {\mathcal {D}}(G)\}\) . The extremal values for a complete multipartite graph are determined, answering a question posed by Xu et al. (Appl Math Comput 433:127414, 2022). The sharp lower and upper bounds for \(\partial _{q}^{+}(D)\) are obtained for graphs G with fixed order and size, where D is any orientation of G. In addition, the sharp lower and upper bounds for \(\partial _{q}^{+}(D)\) are obtained for a graph G with fixed order and connectivity, where D is any orientation of G.
