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On the first Zagreb index of digraphs

Author

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  • Xuan, Hongcong
  • Wang, Shaocong
  • Wang, Jianfeng
  • Li, Yuxia

Abstract

The first Zagreb index Zg+(D) of a digraph D is defined as the sum of squares of out-degrees over all vertices, a generalization of this index based on undirected graphs. In this study, we provide the upper and lower bounds on Zg+(D) whose corresponding extremal digraphs are characterized. We consider the Nordhaus-Gaddum type results for Zg+(D) and obtain the sharp bounds of Zg+(D)+Zg+(D‾). In addition, we derive an expression for Zg+(D)−Zg+(D‾), independent of the structure of D. For the oriented trees D, we determine the maximum and minimum of Zg+(D) and give the orientations of extremal trees. Finally, for fixed number of vertices and arcs, we characterize the local structure of an extremal digraph D maximizing the first Zagreb index and investigate the properties of its adjacency matrix AD and the transpose AD⊤.

Suggested Citation

  • Xuan, Hongcong & Wang, Shaocong & Wang, Jianfeng & Li, Yuxia, 2026. "On the first Zagreb index of digraphs," Applied Mathematics and Computation, Elsevier, vol. 527(C).
    • Handle: RePEc:eee:apmaco:v:527:y:2026:i:c:s009630032600161x
    DOI: 10.1016/j.amc.2026.130109
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