For a connected graph \(\Gamma \) with order n, size m and diameter d, the distance signless Laplacian matrix \(\mathcal {D}^{\mathcal {Q}}(\Gamma )\) is defined as \(\mathcal {D}^{\mathcal {Q}}(\Gamma )={\textit{Tr}}(\Gamma )+\mathcal {D}(\Gamma )\) , where \({\textit{Tr}}(\Gamma )\) is the diagonal matrix of vertex transmissions and \(\mathcal {D}(\Gamma )\) is the distance matrix of \(\Gamma \) . The eigenvalues of \(\mathcal {D}^{\mathcal {Q}}(\Gamma )\) are the distance signless Laplacian eigenvalues of \(\Gamma \) and are denoted by \(\partial _{1}\ge \partial _{2}\ge \dots \ge \partial _{n}\) . The largest eigenvalue \(\partial _1\) is called the distance signless Laplacian spectral radius. Let \(M_k (\Gamma )= \sum _{i=1}^{k}\partial _{i}\) and \(N_k (\Gamma )= \sum _{i=0}^{k-1}\partial _{n-i}\) be the sum of k-largest and the sum of k-smallest distance signless Laplacian eigenvalues of \(\Gamma \) , respectively. In this paper, we obtain the upper bounds for \(M_k (\Gamma )= \sum _{i=1}^{k}\partial _{i}\) and determine the extremal cases. Also, we obtain the upper bounds for \(\partial _1\) and determine the extremal graphs. As a consequence, we obtain the lower bounds for \(N_k (\Gamma )= \sum _{i=0}^{k-1}\partial _{n-i}\) and for smallest eigenvalue \(\partial _n\) 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.
