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

Vertex weighted Laplacian graph energy and other topological indices

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

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

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.

[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.