Vertex weighted Laplacian graph energy and other topological indices

Let $G$ be a graph with a vertex weight $ω$ and the vertices $v_1,\ldots,v_n$. The Laplacian matrix of $G$ with respect to $ω$ is defined as $L_ω(G)=\mathrm{diag}(ω(v_1),\cdots,ω(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $μ_1,\cdots,μ_n$ be eigenvalues of $L_ω(G)$. Then the Laplacian energy of $G$ with respect to $ω$ defined as $LE_ω(G)=\sum_{i=1}^n\big|μ_i - \barω\big|$, where $\barω$ is the average of $ω$, i.e., $\barω=\dfrac{\sum_{i=1}^{n}ω(v_i)}{n}$. 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).

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