On the spectral radius of extended adjacency matrix of a digraph

Let D be a digraph of order n and with a arcs. The degree d i of a vertex v i in D is defined as d i = d (v i) = d i + + d i − , where d i + is the out-degree and d i − is the in-degree of the vertex v i . The extended adjacency matrix A e x (D) of D is an n × n matrix defined as A e x (D) = (b i j) , where b i j = 1 2 (d i d j + d j d i) , whenever (v i , v j) is an arc in D and zero, otherwise. Among the eigenvalues of A e x (D) , the eigenvalue with the largest modulus is the extended adjacency spectral radius or the A e x -spectral radius of D . The contribution of this paper is a series of upper and lower bounds for the A e x -spectral radius in terms of the number of vertices n , the number of arcs a , the vertex degrees, the average degrees of a vertex and the number of closed walks of length 2. The digraphs that are extremal for these bounds are completely characterized. Further, as a consequence, we obtain the corresponding bounds for the extended adjacency spectral radius of a graph. For graphs, our lower bounds improve some known lower bounds for the extended adjacency spectral radius. We compare the different bounds obtained for some classes of digraphs to highlight their importance.