On the sum of distance signless Laplacian eigenvalues of graphs

For a connected graph Γ with order n, size m and diameter d, the distance signless Laplacian matrix DQ(Γ) is defined as DQ(Γ)=Tr(Γ)+D(Γ), where Tr(Γ) is the diagonal matrix of vertex transmissions and D(Γ) is the distance matrix of Γ. The eigenvalues of DQ(Γ) are the distance signless Laplacian eigenvalues of Γ and are denoted by ∂1≥∂2≥⋯≥∂n. The largest eigenvalue ∂1 is called the distance signless Laplacian spectral radius. Let Mk(Γ)=∑i=1k∂i and Nk(Γ)=∑i=0k-1∂n-i be the sum of k-largest and the sum of k-smallest distance signless Laplacian eigenvalues of Γ, respectively. In this paper, we obtain the upper bounds for Mk(Γ)=∑i=1k∂i and determine the extremal cases. Also, we obtain the upper bounds for ∂1 and determine the extremal graphs. As a consequence, we obtain the lower bounds for Nk(Γ)=∑i=0k-1∂n-i and for smallest eigenvalue ∂n and determine the extremal graphs. Moreover, we obtain the upper bounds for the sum of the squares of the vertex transmissions and sum of the squares of the distances of the vertices and show that the bounds are best possible in each case. As an application, we obtain the upper bounds for the distance signless Laplacian energy of graphs and determine the extremal cases.