# Vertex weighted Laplacian graph energy and other topological indices

Document Type: Original Article

Authors

1 Persian Gulf University

2 Department of Mathematics, Faculty of Science, Persian Gulf University, Bushehr 7516913817, I. R. Iran

Abstract

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.

Graphical Abstract

Keywords

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### History

• Receive Date: 03 August 2016
• Revise Date: 01 September 2016
• Accept Date: 01 September 2016