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The outdegree power of oriented graphs

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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.

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Acknowledgements

We are grateful to the reviewers whose comments have helped to considerably improve the presentation of the paper.

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Correspondence to Baoyindureng Wu.

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The work was supported by Open project of Key Laboratory in Xinjiang Uygur Autonomous Region of China (2023D04026) and by NSFC (No. 12061073).

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Ren, Y., Wu, B. The outdegree power of oriented graphs. Comp. Appl. Math. 44, 37 (2025). https://doi.org/10.1007/s40314-024-03001-0

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