The outdegree power of oriented graphs.

For a real number q > 0 , the q-th outdegree power of a digraph D is ∂ q + (D) = ∑ v ∈ V (G) (d D + (v)) q . For a graph G, D (G) is the set of all orientations of G. We focus on a fundamental problem on deciding min { ∂ q + (D) : D ∈ D (G) } and max { ∂ q + (D) : D ∈ 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 ∂ 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 ∂ q + (D) are obtained for a graph G with fixed order and connectivity, where D is any orientation of G. [ABSTRACT FROM AUTHOR]