Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 499-511, 2023


On the signless Laplacian and normalized signless Laplacian spreads of graphs

Emina Milovanović, Serife B. Bozkurt Altindağ, Marjan Matejić, Igor Milovanović

Received January 5, 2022.   Published online February 6, 2023.

Abstract:  Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\geq d_2\geq\cdots\geq d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^+=D+A$ and the normalized signless Laplacian matrix as $\mathcal{L}^+=D^{-1/2}L^+ D^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r(G)= \gamma_2^+/ \gamma_n^+$ and $l(G)=\gamma_2^+-\gamma_n^+$, where $\gamma_1^+ \ge\gamma_2^+\ge\cdots\ge\gamma_n^+ \ge0$ are eigenvalues of $\mathcal{L}^+$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.
Keywords:  Laplacian graph spectra; bipartite graph; spread of graph
Classification MSC:  15A18, 05C50


References:
[1] E. Andrade, G. Dahl, L. Leal, M. Robbiano: New bounds for the signless Laplacian spread. Linear Algebra Appl. 566 (2019), 98-120. DOI 10.1016/j.laa.2018.12.019 | MR 3896162 | Zbl 1410.05114
[2] E. Andrade, M. A. A. de Freitas, M. Robbiano, J. Rodríguez: New lower bounds for the Randić spread. Linear Algebra Appl. 544 (2018), 254-272. DOI 10.1016/j.laa.2017.07.037 | MR 3765785 | Zbl 1388.05108
[3] M. Biernacki, H. Pidek, C. Ryll-Nardzewski: Sur une inéqualité entre des intégrales definies. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 4 (1950), 1-4. (In French.) MR 0042474 | Zbl 0040.31904
[4] Ş. B. Bozkurt Altındağ: Note on the sum of powers of normalized signless Laplacian eigenvalues of graphs. Math. Interdisc. Research 4 (2019), 171-182. DOI 10.22052/mir.2019.208991.1180
[5] 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 (2021), 135-146. DOI 10.7251/BIMVI2101135A | MR 4187056 | Zbl 07540020
[6] Ş. B. Bozkurt, A. D. Güngör, I. Gutman, A. S. Çevik: Randić matrix and Randić energy. MATCH Commun. Math. Comput. Chem. 64 (2010), 239-250. MR 2677585 | Zbl 1265.05113
[7] S. K. Butler: Eigenvalues and Structures of Graphs: Ph.D. Thesis. University of California, San Diego (2008). MR 2711548
[8] M. Cavers, S. Fallat, S. Kirkland: On the normalized Laplacian energy and general Randić index $R_{-1}$ of graphs. Linear Algebra Appl. 433 (2010), 172-190. DOI 10.1016/j.laa.2010.02.002 | MR 2645076 | Zbl 1217.05138
[9] B. Cheng, B. Liu: The normalized incidence energy of a graph. Linear Algebra Appl. 438 (2013), 4510-4519. DOI 10.1016/j.laa.2013.01.003 | MR 3034547 | Zbl 1282.05104
[10] F. R. K. Chung: Spectral Graph Theory. Regional Conference Series in Mathematics 92. AMS, Providence (1997). DOI 10.1090/cbms/092 | MR 1421568 | Zbl 0867.05046
[11] V. Cirtoaje: The best lower bound depended on two fixed variables for Jensen's inequality with ordered variables. J. Inequal. Appl. 2010 (2010), Article ID 128258, 12 pages. DOI 10.1155/2010/128258 | MR 2749168 | Zbl 1204.26031
[12] D. M. Cvetković, M. Doob, H. Sachs: Spectra of Graphs: Theory and Applications. Pure and Applied Mathematics 87. Academic Press, New York (1980). MR 0572262
[13] D. Cvetković, P. Rowlinson, S. K. Simić: Signless Laplacian of finite graphs. Linear Algebra Appl. 423 (2007), 155-171. DOI 10.1016/j.laa.2007.01.009 | MR 2312332 | Zbl 1113.05061
[14] D. Cvetković, S. K. Simić: Towards a spectral theory of graphs based on the signless Laplacian. II. Linear Algebra Appl. 432 (2010), 2257-2277. DOI 10.1016/j.laa.2009.05.020 | MR 2599858 | Zbl 1218.05089
[15] K. C. Das, A. D. Güngör, Ş. B. Bozkurt: On the normalized Laplacian eigenvalues of graphs. Ars Comb. 118 (2015), 143-154. MR 3330443 | Zbl 1349.05205
[16] H. Gomes, I. Gutman, E. Andrade Martins, M. Robbiano, B. San Martín: On Randić spread. MATCH Commun. Math. Comput. Chem. 72 (2014), 249-266. MR 3241719 | Zbl 1464.05070
[17] H. Gomes, E. Martins, M. Robbiano, B. San Martín: Upper bounds for Randić spread. MATCH Commun. Math. Comput. Chem. 72 (2014), 267-278. MR 3241720 | Zbl 1464.05236
[18] R. Gu, F. Huang, X. Li: Randić incidence energy of graphs. Trans. Comb. 3 (2014), 1-9. DOI 10.22108/TOC.2014.5573 | MR 3239628 | Zbl 1463.05331
[19] I. Gutman, E. Milovanović, I. Milovanović: Bounds for Laplacian-type graph energies. Miskolc Math. Notes 16 (2015), 195-203. DOI 10.18514/MMN.2015.1140 | MR 3384599 | Zbl 1340.05164
[20] I. Gutman, N. Trinajstić: Graph theory and molecular orbitals: Total $\phi$-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17 (1972), 535-538. DOI 10.1016/0009-2614(72)85099-1
[21] B. Liu, Y. Huang, J. Feng: A note on the Randić spectral radius. MATCH Commun. Math. Comput. Chem. 68 (2012), 913-916. MR 3052189 | Zbl 1289.05133
[22] M. Liu, B. Liu: The signless Laplacian spread. Linear Algebra Appl. 432 (2010), 505-514. DOI 10.1016/j.laa.2009.08.025 | MR 2577696 | Zbl 1206.05064
[23] A. D. Maden Güngör, A. S. Çevik, N. Habibi: New bounds for the spread of the signless Laplacian spectrum. Math. Inequal. Appl. 17 (2014), 283-294. DOI 10.7153/mia-17-23 | MR 3220994 | Zbl 1408.05082
[24] I. Milovanović, E. Milovanović, E. Glogić: On applications of Andrica-Badea and Nagy inequalities in spectral graph theory. Stud. Univ. Babeş-Bolyai, Math. 60 (2015), 603-609. MR 3437422 | Zbl 1389.05104
[25] D. S. Mitrinović: Analytic Inequalities. Die Grundlehren der mathematischen Wissenschaften 165. Springer, Berlin (1970). DOI 10.1007/978-3-642-99970-3 | MR 274686 | Zbl 0199.38101
[26] M. Randić: Characterization of molecular branching. J. Am. Chem. Soc. 97 (1975), 6609-6615. DOI 10.1021/ja00856a001
[27] L. Shi: Bounds on Randić indices. Discrete Math. 309 (2009), 5238-5241. DOI 10.1016/j.disc.2009.03.036 | MR 2548924 | Zbl 1179.05039
[28] P. Zumstein: Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph: Diploma Thesis. ETH Zürich, Zürich (2005).

Affiliations:   Emina Milovanović (corresponding author), University of Niš, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, Niš 18106, Serbia, e-mail: ema@elfak.ni.ac.rs; Şerife Burcu Bozkurt Altındağ, Karamanoğlu Mehmetbey University, Kamil Özdağ Science Faculty, Department of Mathematics, Karaman, Turkey, e-mail: bozkurtaltindag@kmu.edu.tr; Marjan Matejić, Igor Milovanović, University of Niš, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, Niš 18106, Serbia, e-mail: marjan.matejic@elfak.ni.ac.rs, igor@elfak.ni.ac.rs


 
PDF available at: