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 beta(G) of the strongly connected digraph G is defined as D beta(G) = beta Tr(G)+(1-beta)D(G), for any real 0 <= beta <= 1. The generalized distance spectral radius of G is the spectral radius of D beta(G). Let mu 1 beta(G),mu 2 beta(G),...,mu n beta(G)mu 1 beta(G),mu 2 beta(G),...,mu n beta(G)$ \mu_1<^>\beta{(G)},\mu_2<^>\beta{(G)},...,\mu_n<^>\beta{(G)}$ be the eigenvalues of D beta(G), the generalized distance energy of the digraph G is ED beta(G)=Sigma i=1n|mu i beta(G)-beta W(G)n|ED beta(G)=Sigma i=1n|mu i beta(G)-beta W(G)n|$ {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.