The A_α spectral moments of digraphs with a given dichromatic number

The A_α-matrix of a digraph G is defined as A_α(G)=αD⁺(G)+(1−α)A(G), where α∈[0,1), D⁺(G) is the diagonal outdegree matrix and A(G) is the adjacency matrix. The k-th A_α spectral moment of G is defined as ∑_i=1 ⁿλ_αi ^k, where λ_αi are the eigenvalues of the A_α-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second A_α spectral moment (also known as the A_α energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third A_α spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs.