Vertex weighted Laplacian graph energy and other topological indices

Let G be a graph with a vertex weight omega and the vertices v_۱,ldots,v_n. The Laplacian matrix of G with respect to omega is defined as L_omega(G)=diag(omega(v_۱),cdots,omega(v_n))-A(G), where A(G) is the adjacency matrix of G. Let mu_۱,cdots,mu_n be eigenvalues of L_omega(G). Then the Laplacian energy of G with respect to omega defined as LE_omega (G)=sum_{i=۱}^nbig|mu_i - overline{omega}big|, where overline{omega} is the average of omega, i.e., overline{omega}=dfrac{sum_{i=۱}^{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).\[۵mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.

Keywords:

energy of graph , Laplacian energy , Vertex weight , Topological index , toroidal fullerenes