Vertex weighted Laplacian graph energy and other topological indices

Abstract

Let $G$ be a graph with a vertex weight $\omega$ and the vertices $v_1,\dots ,v_n$. 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 $L{E}_{\omega }(G)={\sum }_{i=1}^n|{\mu }_i-\bar{\omega }|$, where $\bar{\omega }$ is the average of $\omega$, i.e., $\bar{\omega }=\frac{{\sum }_{i=1}^n\omega (v_i)}{n}$. 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).