Vertex weighted Laplacian graph energy and other topological indices

Document Type : Original Article


1 Persian Gulf University

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


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

Vertex weighted Laplacian graph energy and other topological indices


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Volume 8, Issue 4
December 2023
Pages 201-209
  • Receive Date: 24 October 2023
  • Revise Date: 05 November 2023
  • Accept Date: 19 November 2023
  • Publish Date: 01 December 2023