normalized distance laplacian energy change due to edge deletion in complete graph and complete multipartite graph

Let g be a connected graph with a distance matrix d. the eigenvalues of d forms the distance spectrum or d−spectrumof g. the transmission of a vertex v in g is the sum of the distances from v to all other vertices of g and t (g) is the diagonal matrix with transmission of the vertices as diagonal entries. the distance laplacian matrix is defined as dl(g)=t(g)−d(g) and the normalized distance laplacian matrix as dl(g)=t (g)−1/2dl(g)t (g)−1/2. the normalized-distance laplacian energy is defined as dle(g)=∑i=1n |ρil−1|. let dle(g − e) be the normalized laplacian energy of distance when an edge e is removed. in this paper we are studying the normalized distance laplacianenergy change due to an edge deletion.