Sharafdini, R., Panahbar, H. (2016). Vertex weighted Laplacian graph energy and other topological indices. Journal of Mathematical Nanoscience, 6(1-2), 57-65. doi: 10.22061/jmns.2016.524

Reza Sharafdini; Habibeh Panahbar. "Vertex weighted Laplacian graph energy and other topological indices". Journal of Mathematical Nanoscience, 6, 1-2, 2016, 57-65. doi: 10.22061/jmns.2016.524

Sharafdini, R., Panahbar, H. (2016). 'Vertex weighted Laplacian graph energy and other topological indices', Journal of Mathematical Nanoscience, 6(1-2), pp. 57-65. doi: 10.22061/jmns.2016.524

Sharafdini, R., Panahbar, H. Vertex weighted Laplacian graph energy and other topological indices. Journal of Mathematical Nanoscience, 2016; 6(1-2): 57-65. doi: 10.22061/jmns.2016.524

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 $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, 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 $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. 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).\[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.