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Milovanović, Emina ; Bozkurt Altındağ, Şerife Burcu ; Matejić, Marjan ; Milovanović, Igor
Inequalities for real number sequences with applications in spectral graph theory. (English). Czechoslovak Mathematical Journal, vol. 72 (2022), issue 3, pp. 783-799
MSC: 05C30, 15A18 | MR 4467942 | Zbl 07584102 | DOI: 10.21136/CMJ.2022.0155-21
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Keywords:
inequality; real number sequence; Laplacian eigenvalue of graph; normalized Laplacian eigenvalue
Summary:
Let $a=(a_{1},a_{2},\ldots ,a_{n})$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,\ldots ,n\}$ the index set and by $J_{k}=\{I= \{ r_{1},r_{2},\ldots ,r_{k} \}$, $1\leq r_{1}<r_{2}< \nobreak \cdots <r_{k}\leq n\}$ the set of all subsets of $S$ of cardinality $k$, $1\leq k\leq n-1$. In addition, denote by $a_{I}=a_{r_{1}}+a_{r_{2}}+\cdots +a_{r_{k}}$, $1\leq k\leq n-1$, $1\leq r_{1}<r_{2}<\cdots <r_{k}\leq n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_{1}}=a_{1}+a_{2}+\cdots +a_{k}$ and $a_{I_{n}}=a_{n-k+1}+a_{n-k+2}+\cdots +a_{n}$. We consider bounds of the quantities $RS_{k}(a)=a_{I_{1}}/a_{I_{n}}$, $LS_{k}(a)=a_{I_{1}}-a_{I_{n}}$ and $S_{k,\alpha }(a)=\sum _{I\in J_{k}}a_{I}^{\alpha }$ in terms of $A=\sum _{i=1}^{n}a_{i}$ and $B=\sum _{i=1}^{n}a_{i}^{2}$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.
References:
[1] Andrade, E., Freitas, M. A. A. de, Robbiano, M., Rodríguez, J.: 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
[2] Bianchi, M., Cornaro, A., Palacios, J. L., Torriero, A.: Bounds for the Kirchhoff index via majorization techniques. J. Math. Chem. 51 (2013), 569-587. DOI 10.1007/s10910-012-0103-x | MR 3017758 | Zbl 1327.05066
[3] Butler, S. K.: Eigenvalues and Structures of Graphs: Ph.D. Thesis. University of California, San Diego (2008). MR 2711548
[4] Cavers, M., Fallat, S., Kirkland, S.: 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
[5] Chen, X., Das, K. C.: Some results on the Laplacian spread of a graph. Linear Algebra Appl. 505 (2016), 245-260. DOI 10.1016/j.laa.2016.05.002 | MR 3506494 | Zbl 1338.05158
[6] Chen, X., Qian, J.: Bounding the sum of powers of the Laplacian eigenvalues of graphs. Appl. Math., Ser. B (Engl. Ed.) 26 (2011), 142-150. DOI 10.1007/s11766-011-2732-4 | MR 2810546 | Zbl 1240.05186
[7] Chung, F. R. K.: Spectral Graph Theory. Regional Conference Series in Mathematics 92. AMS, Providence (1997). DOI 10.1090/cbms/092 | MR 1421568 | Zbl 0867.05046
[8] Edwards, C. S.: The largest vertex degree sum for a triangle in a graph. Bull. Lond. Math. Soc. 9 (1977), 203-208. DOI 10.1112/blms/9.2.203 | MR 0463005 | Zbl 0357.05058
[9] Fath-Tabar, G. H., Ashrafi, A. R.: Some remarks on Laplacian eigenvalues and Laplacian energy of graphs. Math. Commun. 15 (2010), 443-451. MR 2814304 | Zbl 1206.05062
[10] Goldberg, F.: Bounding the gap between extremal Laplacian eigenvalues of graphs. Linear Algebra Appl. 416 (2006), 68-74. DOI 10.1016/j.laa.2005.07.007 | MR 2232920 | Zbl 1107.05059
[11] Grone, R., Merris, R.: The Laplacian spectrum of graph. II. SIAM J. Discrete Math. 7 (1994), 221-229. DOI 10.1137/S0895480191222653 | MR 1271994 | Zbl 0795.05092
[12] Gutman, I., Trinajstić, N.: 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
[13] Hakimi-Nezhaad, M., Ashrafi, A. R.: A note on normalized Laplacian energy of graphs. J. Contemp. Math. Anal., Armen. Acad. Sci. 49 (2014), 207-211. DOI 10.3103/S106836231405001X | MR 3379554 | Zbl 1312.05082
[14] Huang, J., Li, S.: On the normalized Laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs. Bul. Aust. Math. Soc. 91 (2015), 353-367. DOI 10.1017/S0004972715000027 | MR 3338961 | Zbl 1326.05082
[15] Jensen, J. L. W. V.: Sur les functions convexes et les inéqualités entre les valeurs moyennes. Acta Math. 30 (1906), 175-193 French \99999JFM99999 37.0422.02. DOI 10.1007/BF02418571 | MR 1555027
[16] Kemeny, J. G., Snell, J. L.: Finite Markov Chains. The University Series in Undergraduate Mathematics. Van Nostrand, Princeton (1960). MR 0115196 | Zbl 0089.13704
[17] Li, J., Guo, J.-M., Shiu, W. C., ndağ, Ş. B. B. Altı, Bozkurt, D.: Bounding the sum of powers of normalized Laplacian eigenvalues of a graph. Appl. Math. Comput. 324 (2018), 82-92. DOI 10.1016/j.amc.2017.12.003 | MR 3743658 | Zbl 1426.05101
[18] Merris, R.: Laplacian matrices of graphs: A survey. Linear Algebra Appl. 197-198 (1994), 143-176. DOI 10.1016/0024-3795(94)90486-3 | MR 1275613 | Zbl 0802.05053
[19] Milovanović, I. Ž., Milovanović, E. I., Glogić, E.: Lower bounds of the Kirchhoff and degree Kirchhoff indices. Sci. Publ. State Univ. Novi Pazar, Ser. A, Appl. Math. Inf. Mech. 7 (2015), 25-31. DOI 10.5937/SPSUNP1501025M
[20] Milovanović, I. Ž., Milovanović, E. I., Glogić, E.: On Laplacian eigenvalues of connected graphs. Czech. Math. J. 65 (2015), 529-535. DOI 10.1007/s10587-015-0191-4 | MR 3360442 | Zbl 1363.15016
[21] Milovanović, I. Ž., Milovanović, E. I.: Bounds for the Kirchhoff and degree Kirchhoff indices. Bounds in Chemical Graph Theory: Mainstreams Mathematical Chemistry Monographs 20. University of Kragujevac, Kragujevac (2017), 93-119. MR 3403904
[22] Mitrinović, D. S., Pečarić, J. E., Fink, A. M.: Classical and New Inequalities in Analysis. Mathematics and Its Applications. East European Series 61. Kluwer Academic Publishers, Dorchrecht (1993). DOI 10.1007/978-94-017-1043-5 | MR 1220224 | Zbl 0771.26009
[23] Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326 (2007), 1472-1475. DOI 10.1016/j.jmaa.2006.03.072 | MR 2280998 | Zbl 1113.15016
[24] Nordhaus, E. A., Gaddum, J. W.: On complementary graphs. Am. Math. Mon. 63 (1956), 175-177. DOI 10.2307/2306658 | MR 0078685 | Zbl 0070.18503
[25] Ozeki, N.: On the estimation of the inequalities by the maximum, or minimum values. J. College Arts Sci. Chiba Univ. 5 (1968), 199-203 Japanese. MR 0254198
[26] Palacios, J. L.: Some inequalities for Laplacian descriptors via majorization. MATCH Commun. Math. Comput. Chem. 77 (2017), 189-194. MR 3645376 | Zbl 1466.92279
[27] Palacios, J. L., Renom, J. M.: Broder and Karlin's formula for hitting times and the Kirchhoff index. Int. J. Quantum Chem. 111 (2011), 35-39. DOI 10.1002/qua.22396
[28] Shi, L.: Bounds on Randić indices. Discr. Math. 309 (2009), 5238-5241. DOI 10.1016/j.disc.2009.03.036 | MR 2548924 | Zbl 1179.05039
[29] Shi, L., Wang, H.: The Laplacian incidence energy of graphs. Linear Algebra Appl. 439 (2013), 4056-4062. DOI 10.1016/j.laa.2013.10.028 | MR 3133474 | Zbl 1282.05152
[30] You, Z., Liu, B.: On the Laplacian spectral ratio of connected graphs. Appl. Math. Lett. 25 (2012), 1245-1250. DOI 10.1016/j.aml.2011.09.071 | MR 2947387 | Zbl 1248.05116
[31] You, Z., Liu, B.: The Laplacian spread of graphs. Czech. Math. J. 62 (2012), 155-168. DOI 10.1007/s10587-012-0003-z | MR 2899742 | Zbl 1245.05089
[32] Zhou, B.: On sum of powers of the Laplacian eigenvalues of graphs. Linear Algebra Appl. 429 (2008), 2239-2246. DOI 10.1016/j.laa.2008.06.023 | MR 2446656 | Zbl 1144.05325
[33] Zumstein, P.: Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph: Diploma Thesis. ETH, Zürich (2005).
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