Spectral norm and energy of a digraph with respect to a VDB topological index

The set of singular values of a digraph with respect to a vertex-degree based topological index is the set of all singular values of its general adjacency matrix. The spectral norm is the largest singular value and the energy the sum of the singular values. In this paper we characterize the digraphs which have exactly one singular value different from zero and the digraphs for which all singular values are equal. As a consequence, we deduce sharp upper and lower bounds for the spectral norm and energy of digraphs. In addition to being a natural generalization, proving the results in the general setting of digraphs allows us to deduce new results on graph energy.