Vertex weighted Laplacian graph energy and other topological indices
  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 $\omega$ is defined 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 eigenvalues of $L_\omega(G)$. 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 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). 
Reza Sharafdini, H. Panahbar
math.CO