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

Vertex weighted Laplacian graph energy and other topological indices

Keywords


[1] A. R. Ashrafi, Wiener Index of Nanotubes, Toroidal Fullerenes and Nanostars, In The Mathemat-ics and Topology of Fullerenes, F. Cataldo, A. Graovac and O. Ori, (Eds.), Springer Netherlands: Dordrecht, 2011, pp. 21–38.
[2] T. Aleksic, Upper bounds for Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 435–439.
[3] F. K. Bell, A Note on the Irregularity of Graphs, Linear Algebra Appl. 161 (1992) 45–54.
[4] M. S. Cavers, The normalized laplacian matrix and general randic index of graphs. Ph.D. Thesis, University of Regina, Regina, Saskatchewan, 2010.
[5] S. Cabello and P. Luksic, The complexity of obtaining a distance-balanced graph, Electron. J. Com-bin. 18 (1) (2011) Paper 49.
[6] K. Ch. Das, S. A. Mojallal and I. Gutman, On energy and Laplacian energy of bipartite graphs, Appl. Math. Comput. 273 (2016) 759–766.
[7] N. N. M. de Abreu, C. M. Vinagre, A. S. Bonifacio and I. Gutman, The Laplacian energy of some Laplacian integral graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 447–460.
[8] R. Grone and R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229.
[9] R. Grone, R. Merris and V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218–238.
[10] I. Gutman, The energy of a Graph, Old and New Results. In Algebraic Combinatorics and Appli-cations A. Betten, A. Kohnert, R. Laue and A. Wassermann (Eds.), Springer-Verlag: Berlin, 2001, pp. 196–211.
[11] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz. 103 (1978) 1–22.
[12] I. Gutman, N.M.M. de Abreu, C.T.M. Vinagre, A.S. Bonifacio and S. Radenkovic, Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem. 59 (2008) 343–354.
[13] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag: Berlin, 1986, Chapter 8.
[14] I. Gutman, S. Zare Firoozabadi, J. A. de la Pena and J. Rada, On the energy of regular graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 435–442.
[15] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37.

[16] I. Gutman and P. Paule, The variance of the vertex degrees of randomly generated graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 30–35.
[17] K. Handa, Bipartite graphs with balanced (a, b)-partitions, Ars Combin. 51 (1999) 113–119.
[18] F. Harary, Status and contrastatus, Sociometry, 22 (1959) 23–43.
[19] G. Indulal and A. Vijayakumar, A note on energy of some graphs, MATCH Commun. Math. Com-put. Chem. 59 (2008) 269–274.
[20] R. Merris, A survey of graph Laplacians, Linear and Multilinear Algebra, 39 (1995) 19–31.
[21] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994) 143–176.
[22] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Wiley, New York, 1991, pp.
871–898.
[23] B. Mohar, Graph Laplacians, in: L. W. Brualdi and R. J. Wilson (Eds.), Topics in. Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 2004, pp. 113–136.
[24] M. Robbiano and R. Jimenez, Applications of a theorem by Ky Fan in the theory of Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 62 (2009) 537–552.
[25] W. So, M. Robbiano, N. M. M. de Abreu and I. Gutman, Applications of the Ky Fan theorem in the theory of graph energy, Linear Algebra Appl. 432 (2010) 2163–2169.
[26] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
[27] R. Sharafdini and H. Panahbar, On Laplacian energy of vertex weighted graphs, manuscript.
[28] R. Sharafdini, A. Ataei and H. Panahbar, Applications of a theorem by Ky Fan in the theory of weighted Laplacian graph energy, Submited, eprint arXiv:1608.07939.
[29] B. Zhou, I. Gutman and T. Aleksic, A note on Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441–446.
[30] S. Yousefi, H. Yousefi-Azari, A. R. Ashrafi and M. H. Khalifeh, Computing Wiener and Szeged Indices of an Achiral Polyhex Nanotorus, JSUT, 33 (3) (2008) 7–11.


Volume 6, Issue 1
Winter and Spring 2016
Pages 57-65
  • Receive Date: 03 August 2016
  • Revise Date: 01 September 2016
  • Accept Date: 01 September 2016