On Distance Laplacian Spread, Wiener Index and Independence Number of a Graph

Let G be a simple connected simple graph of order n. The distance Laplacian matrix DL(G) is defined as DL(G)=Diag(Tr)-D(G) , where Diag(Tr) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The eigenvalues of DL(G) are the distance Laplacian eigenvalues of G and are denoted by ∂ ₁L(G), ∂ ₂L(G), ,∂ ₙL(G) . The distance Laplacian spread DLS(G) of a connected graph G is the difference between largest and second smallest distance Laplacian eigenvalues, that is, ∂ ₁L(G)-∂ ₙ₋₁L(G) . We obtain bounds for DLS(G) in terms of the Wiener index W(G), order n and the maximum transmission degree Trₘₐₓ(G) of G and characterize the extremal graphs. We obtain two lower bounds for DLS(G), the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected k-partite graph G, k≤ n-1 , with n vertices having disconnected complement, we show that DLS(G)≥ n/k with equality if and only if G is a complete balanced k-partite graph.