Bounds for the $\boldsymbol{A_{\alpha}}$-spectral radius of a digraph

H. A. Ganie, M. Baghipur

Let $ D$ be a digraph of order $n$ and let $ A(D) $ be the adjacency matrix of $D$. Let $ Deg(D) $ be the diagonal matrix of vertex out-degrees of $ D$. For any real $ \alpha\in [0,1], $ the generalized adjacency matrix $ A_{\alpha}(D) $ of the $D$ is defined as $ A_{\alpha}(D)=\alpha Deg(D)+(1-\alpha)A(D).$ The largest modulus of the eigenvalues of $ A_{\alpha}(D) $ is called the generalized adjacency spectral radius or the $ A_{\alpha} $-spectral radius of $ D$. In this paper, we obtain some new upper and lower bounds for the spectral radius of $ A_{\alpha}(D) $ in terms of the number of vertices $n$, the number of arcs, the vertex out-degrees, the average 2-out-degrees of the vertices of $ D $ and the parameter~$ \alpha $. We characterize the extremal digraphs attaining these bounds.