The A_α spectral moments of digraphs with a given dichromatic number

The Aα-matrix of a digraph G is defined as Aα(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 Aα spectral moment of G is defined as ∑i=1 nλαi k, where λαi are the eigenvalues of the Aα-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second Aα spectral moment (also known as the Aα energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third Aα 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.</p

Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number

The k-th Laplacian spectral moment of a digraph G is defined as ∑i=1nλik, where λi are the eigenvalues of the Laplacian matrix of G and k is a nonnegative integer. For k=2, this invariant is better known as the Laplacian energy of G. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for k≥3 as an open problem.</p