On the eigenvalues of the distance signless Laplacian matrix of graphs.

Let G be a connected graph and let DQ(G) be the distance signless Laplacian matrix of G with eigenvalues ρ1 ≥ ρ2 ≥ · · · ≥ ρn. The spread of the matrix DQ(G) is defined as s(DQ(G)) := maxi,j |ρi - ρj | = ρ1 - ρn. We derive new bounds for the distance signless Laplacian spectral radius ρ1 of G. We establish a relationship between the distance signless Laplacian energy and the spread of DQ(G). For a real number α ≠ 0, the graph invariant mα(G) is the sum of the α-th power of the distance signless Laplacian eigenvalues of G. Finally, we obtain various bounds for the graph invariant mα(G). [ABSTRACT FROM AUTHOR]