Sharp bounds on the generalized distance spectral radius and generalized distance energy of strongly connected digraphs

Let D (G) be the distance matrix of a strongly connected digraph G, Tr (G) be the diagonal matrix with vertex transmissions of G as diagonal entries. The generalized distance matrix D_ (G) of the strongly connected digraph G is defined as D_ (G) = Tr (G) + (1-) D (G), for any real 0 1. The generalized distance spectral radius of G is the spectral radius of D_ (G). Let ^₁ (G), ^₂ (G), , ^ₙ (G) be the eigenvalues of D_ (G), the generalized distance energy of the digraph G is E₃_ (G) =₈=₁ⁿ|^ᵢ (G) - W (G) n|, where W (G) is the sum of distances between all ordered pairs of vertices of G. In this paper, we obtain some sharp upper and lower bounds for the generalized distance spectral radius of G and characterize the extremal digraphs. Moreover, we also give some lower bounds on the generalized distance energy of digraphs.