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_β(G) of the strongly connected digraph G is defined as D_β(G)=β Tr(G)+(1-β)D(G), for any real 0≤β≤ 1. The generalized distance spectral radius of G is the spectral radius of D_β(G). Let μ^β₁(G),μ^β₂(G),…,μ^βₙ(G) be the eigenvalues of D_β(G), the generalized distance energy of the digraph G is ED_β(G)=∑ᵢ₌₁ⁿ|μ^βᵢ(G)-β 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.