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

\nLet 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 Dβ(G) of the strongly connected digraph G is defined as Dβ(G) = βTr(G)+(1−β)D(G), for any real 0 ≤ β ≤ 1. The generalized distance spectral radius of G is the spectral radius of Dβ(G). Let $ \\mu_1^\\beta{(G)},\\mu_2^\\beta{(G)},...,\\mu_n^\\beta{(G)}$ be the eigenvalues of Dβ(G), the generalized distance energy of the digraph G is $ {E_D}_\\beta{(G)}=\\overset n{\\underset{i=1}{\\mathrm\\Sigma}}{\\vert\\mu_i^\\beta{(G)}-\\frac{\\beta W{(G)}}n\\vert}$, 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.\n