On the generalized adjacency spectral radius of digraphs. Issue 18 (12th December 2022) - Ganie, Hilal A.; Baghipur, Maryam
ABSTRACT: Let D be a digraph of order n and let A ( D ) be the adjacency matrix of D . Let D e g ( D ) be the diagonal matrix of vertex out-degrees of D . For any real α ∈ [ 0, 1 ], the generalized adjacency matrix A α ( D ) of the digraph D is defined as A α ( D ) = α D e g ( D ) + ( 1 − α ) A ( D ) . The largest modulus of the eigenvalues of A α ( D ) is called the generalized adjacency spectral radius or the A α -spectral radius of D . In this paper, we obtain some lower bounds for the spectral radius of A α ( D ) in terms of the number of vertices, the number of arcs and the number of closed walks of the digraph D . The extremal graphs attained these lower bounds are determined. We also obtain some bounds for the spectral radius of A α ( D ) in terms of the vertex out-degrees, the vertex average 2-out-degrees of the vertices of D and parameter α . We characterize the extremal digraphs attaining these bounds.