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Let $G$ be a graph with a vertex weight $\omega$ and the vertices$v_1,\ldots,v_n$. The Laplacian matrix of $G$ with respect to $\omega$ isdefined as  $L_\omega(G)=\mathrm{diag}(\omega(v_1),\cdots,\omega(v_n))-A(G)$, where$A(G)$ is the adjacency matrix of $G$. Let $\mu_1,\cdots,\mu_n$ be eigenvaluesof $L_\omega(G)$. Then the Laplacian energy of $G$ with respect to $\omega$defined as $LE_\omega (G)=\sum_{i=1}^n\big|\mu_i - \bar{\omega}\big|$, where$\bar{\omega}$ is the average of $\omega$, i.e.,$\bar{\omega}=\dfrac{\sum_{i=1}^{n}\omega(v_i)}{n}$. In this paper we considerseveral natural vertex weights of $G$ and obtain some inequalities between theordinary and Laplacian energies of $G$ with corresponding vertex weights.Finally, we apply our results to the molecular graph of toroidal fullerenes (orachiral polyhex nanotorus).

Vertex weighted Laplacian graph energy and other topological indices