The maximum outdegree power of complete k-partite oriented graphs

•In this paper, we investigate the outdegree power of oriented graphs, which is a generalization of degree power of simple graphs.•We apply a very interesting and efficient tool, that is, majorization in our proof of main result. Suppose that Gσ is an oriented graph with order n and orientation σ. For d1+≥d2+≥⋯≥dn+, the outdegree sequence of Gσ is denoted by (d1+,d2+,…,dn+). Similar to the definition of the degree power for a simple graph, define the outdegree power for an oriented graph Gσ by ∂q+(Gσ)=∑i=1n(di+)q where q is a positive integer. Obviously, ∂1+(Gσ)=|E(G)|. Denote by G(G) the set of oriented graphs whose underlying graph is isomorphic to G, where G is a simple graph. In the paper, we concentrate on the problem: How large the value of ∂q+(Gσ) could be among all oriented graphs in G(G)? Using majorization, we prove that Kn1,…,nk→ is the unique extremal graph which attains the maximum outdegree power ∂q+(Gσ) over all oriented graphs in G(Kn1,…,nk) for q>ln(n−n1+1)ln(1+1n−n1−1). Further, we present several results about the extrema of ∂q+(Gσ) over all complete k-partite oriented graphs.