For a digraph \({\mathcal {D}}\) with n vertices, a arcs and the outdegree sequence \(d_1^{+}, d_2^{+},\dots , d_n^{+}\) of vertices of \({\mathcal {D}}\) . The first outdegree Zagreb index of \({\mathcal {D}}\) is \(Zg^{+}({\mathcal {D}})\) , which is defined as \(Zg^{+}({\mathcal {D}})=\sum \limits _{i=1}^{n}(d_i^{+})^2\) . This work establishes new upper and lower bounds for the first outdegree Zagreb index \(Zg^{+}({\mathcal {D}})\) of a digraph \({\mathcal {D}}\) , expressed in terms of various structural invariants. The digraphs that achieve these extremal bounds are fully characterized. In particular, we investigate the problem of determining the orientations that maximize or minimize the first outdegree Zagreb index for the wheel graph \(W_n\) .
