On the Aα spectral radius of generalized weighted digraphs

Let G = (V(G), E(G)) be a generalized weighted digraph without loops and multiple arcs, where the weight of each arc is a nonnegative and symmetric matrix of same order p. For vi ? V(G), let w+i = ?vj?N+i wij, where wij is the weight of the arc (vi, vj), and N+i is the set of out-neighbors of the vertex vi. Let A? (G) = ?D(G) + (1-?)A(G), where 0 ? ? ? 1, A(G) is the adjacency matrix of the generalized weighted digraph G, and D(G) = diag(w+1, w+2,..., w+n). The spectral radius of A? (G) is called the A? spectral radius of G. In this paper, we give some upper bounds on the A? spectral radius of generalized weighted digraphs, and characterize the digraphs achieving the upper bounds. As application, we obtain some upper bounds on the A? spectral radius of weighted digraphs and unweighted digraphs.