Vertex weighted Laplacian graph energy and other topological indices

Let g be a graph with a vertex weight ω and the vertices v_1, . . . ,v_n. the laplacian matrix of g with respect to ω is defined as l_ω(g) = diag(ω(v_1), · · · ,ω(v_n)) − a(g), where a(g) is the adjacency matrix of g. let μ_1, · · · ,μ_n be eigenvalues of l_ω(g). then the laplacian energy of g with respect to ω is defined as le_ω(g) = sum_{i=1}^nbig|mu_i - overline{omega}big|$ , where ϖ is the average of ω, i.e., ϖ = $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. in this paper, we consider several natural vertex weights of g and obtain some inequalities between the ordinary and laplacian energies of g with corresponding vertex weights. finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).