The outdegree power of oriented graphs

Abstract

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.