On graphs with minimal distance signless Laplacian energy

For a simple connected graph G of order n having distance signless Laplacian eigenvalues , the distance signless Laplacian energy DSLE(G) is defined as where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph K ∨ ( K ∪ K ), has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.