On Normalized Signless Laplacian Resolvent Energy


Download PDF

Authors: S. B. BOZKURT ALTINDAG, I. MILOVANOVIć, E. MILOVANOVIć AND M. MATEJIć

DOI: 10.46793/KgJMat2405.673A

Abstract:

Let G be a simple connected graph with n vertices. Denote by +(G ) = D(G ) 12Q(G  ) D(G ) 12 the normalized signless Laplacian matrix of graph G, where Q(G ) and D(G ) are the signless Laplacian and diagonal degree matrices of G, respectively. The eigenvalues of matrix +(G), 2 = γ1+ γ2+ ⋅ ⋅⋅γn+ 0, are normalized signless Laplacian eigenvalues of G. In this paper, we introduce the normalized signless Laplacian resolvent energy of G as ERNS(G ) = i=1n3−1γ+
    i. We also obtain some lower and upper bounds for ERNS(G ) as well as its relationships with other energies and signless Kemeny’s constant.



Keywords:

Normalized signless Laplacian eigenvalues, normalized signless Laplacian resolvent energy, bounds.



References:

[1]   L. E. Allem, J. Capaverde, V. Trevisan, I. Gutman, E. Zogić and E. Glogić, Resolvent energy of unicyclic, bicyclic and tricyclic graphs, MATCH Commun. Math. Comput. Chem. 77 (2017), 95-104.

[2]   Ş. B. Bozkurt, A. D. Gungor, I. Gutman and A. S. Cevik, Randić matrix and Randić energy, MATCH Commum. Math. Comput. Chem. 64 (2010), 239–250.

[3]   Ş. B. Bozkurt Altındağ, Note on the sum of powers of normalized signless Laplacian eigenvalues of graphs, Mathematics Interdisciplinary Research 4(2) (2019), 171–182. https://dx.doi.org/10.22052/mir.2019.208991.1180

[4]   Ş. B. Bozkurt Altındağ, Sum of powers of normalized signless Laplacian eigenvalues and Randić (normalized) incidence energy of graphs, Bull. Int. Math. Virtual Inst. 11(1) (2021), 135–146. https://doi.org/10.7251/BIMVI2101135A

[5]   Ş. B. Bozkurt Altındağ, I. Milovanović, M. Matejić and E. Milovanović, On the degree Kirchhoff index of bipartite graphs, Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and Mechanics 13(1) (2021), 1–8.

[6]   S. Butler, Algebraic aspects of the normalized Laplacian, in: A. Beveridge, J. Griggs , L. Hogben, G. Musiker and P. Tetali (Eds.), Recent Trends in Combinatorics, IMA Vol. Math. Appl. 159, Springer, 2016, 295–315. https://doi.org/10.1007/978-3-319-24298-9\_13

[7]   A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastine, M. D. Safe, V. Trevisan and I. Gutman, Some results for the (signless) Laplacian resolvent, MATCH Commun. Math. Comput. Chem. 77 (2017), 105–114.

[8]   M. Cavers, S. Fallat and S. Kirkland, On the normalized Laplacian energy and general Randić index R1 of graphs, Linear Algebra Appl. 433 (2010), 172–190. https://doi.org/10.1016/j.laa.2010.02.002

[9]   B. Cheng and B. Liu, The normalized incidence energy of a graph, Linear Algebra Appl. 438 (2013), 4510–4519. https://doi.org/10.1016/j.laa.2013.01.003

[10]   F. R. K. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, 1997.

[11]   D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs, Academic Press, New York, 1980.

[12]   K. Ch. Das, A. D. Gungor and Ş. B. Bozkurt, On the normalized Laplacian eigenvalues of graphs, Ars Combin. 118 (2015), 143–154.

[13]   M. Ghebleh, A. Kanso and D. Stevanović, On trees with smallest resolvent energy, MATCH Commun. Math. Comput. Chem. 77 (2017), 635-654.

[14]   R. Gu, F. Huang and X. Li, Randić incidence energy of graphs, Trans. Comb. 3(4) (2014), 1–9. https://doi.org/10.22108/TOC.2014.5573

[15]   I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz Graz. 103 (1978), 1–22.

[16]   I. Gutman, D. Kiani and M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009), 573–580.

[17]   I. Gutman, B. Furtula and Ş. B. Bozkurt, On Randić energy, Linear Algebra Appl. 442 (2014), 50–57. https://doi.org/10.1016/j.laa.2013.06.010

[18]   I. Gutman, B. Furtula, E. Zogić and E. Glogić, Resolvent energy of graphs, MATCH Commun. Math. Comput. Chem. 75 (2016), 279-290.

[19]   I. Gutman and X. Li, Graph Energies - Theory and Applications, University of Kragujevac, Kragujevac, 2016.

[20]   P. Henrići, Two remarks on the Kantorovich inequality, Amer. Math. Monthly 68 (1961), 904-906. https://doi.org/10.2307/2311698

[21]   M. Levene and G. Loizou, Kemeny’s constant and the random surfer, Amer. Math. Monthly 109 (2002), 741–745. https://doi.org/10.2307/3072398

[22]   X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.

[23]   J. Li, J. M. Guo and W. C. Shiu, Bounds on normalized Laplacian eigenvalues of graphs, J. Inequal. Appl. 316 (2014), 1–8. https://doi.org/10.1186/1029-242X-2014-316

[24]   B. Liu, Y. Huang and J. Feng, A note on the Randić spectral radius, MATCH Commun. Math. Comput. Chem. 68 (2012), 913–916.

[25]   J. Liu and B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008), 355–372.

[26]   R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994), 143–176. https://doi.org/10.1016/0024-3795(94)90486-3

[27]   E. I. Milovanović, M. M. Matejić and I. Ž Milovanović, On the normalized Laplacian spectral radius, Laplacian incidence energy and Kemeny’s constant, Linear Algebra Appl. 582 (2019), 181–196. https://doi.org/10.1016/j.laa.2019.08.004

[28]   I. Milovanović, E. Milovanović, M. Matejić and Ş. B. Bozkurt Altındağ, Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs, Commn. Comb. Optim. 6(2) (2021), 259–271. https://doi.org/22049/CCO.2021.26987.1173

[29]   D. S. Mitrinović and P. M. Vasić, Analytic Inequalities, Springer Verlag, Berlin, Heidelberg, New York, 1970.

[30]   J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, Wien, Hölder, 1913, 1295–1438.

[31]   M. Randić, On characterization of molecule branching, J. Amer. Chem. Soc. 97 (1975), 6609–6615. https://pubs.acs.org/doi/10.1021/ja00856a001

[32]   L. Shi and H. Wang, The Laplacian incidence energy of graphs, Linear Algebra Appl. 439 (2013), 4056–4062. https://doi.org/10.1016/j.laa.2013.10.028

[33]   S. Sun and K. C. Das, Comparison of resolvent energies of Laplacian matrices, MATCH Commun. Math. Comput. Chem. 82 (2019), 491–514.

[34]   E. Zogić and E. Glogić, New bounds for the resolvent energy of graphs, Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and Mechanics 9(2) (2017), 187–191. https://doi.org/10.5937/SPSUNP1702187Z

[35]   E. Zogić, B. Borovićanin, E. Glogić, I. Milovanović and E. Milovanović, New bounds for some spectrum-based topological indices of graphs, MATCH Commun. Math. Comput. Chem. 86 (2021), 685–701.

[36]   P. Zumstein, Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph, Th. Diploma, ETH Zürich, 2005.