| Title:
|
Inequalities for real number sequences with applications in spectral graph theory (English) |
| Author:
|
Milovanović, Emina |
| Author:
|
Bozkurt Altındağ, Şerife Burcu |
| Author:
|
Matejić, Marjan |
| Author:
|
Milovanović, Igor |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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72 |
| Issue:
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3 |
| Year:
|
2022 |
| Pages:
|
783-799 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
inequality |
| Keyword:
|
real number sequence |
| Keyword:
|
Laplacian eigenvalue of graph |
| Keyword:
|
normalized Laplacian eigenvalue |
| MSC:
|
05C30 |
| MSC:
|
15A18 |
| idZBL:
|
Zbl 07584102 |
| idMR:
|
MR4467942 |
| DOI:
|
10.21136/CMJ.2022.0155-21 |
| . |
| Date available:
|
2022-08-22T08:23:07Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150617 |
| . |
| Reference:
|
[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. Zbl 1388.05108, MR 3765785, 10.1016/j.laa.2017.07.037 |
| Reference:
|
[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. Zbl 1327.05066, MR 3017758, 10.1007/s10910-012-0103-x |
| Reference:
|
[3] Butler, S. K.: Eigenvalues and Structures of Graphs: Ph.D. Thesis.University of California, San Diego (2008). MR 2711548 |
| Reference:
|
[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. Zbl 1217.05138, MR 2645076, 10.1016/j.laa.2010.02.002 |
| Reference:
|
[5] Chen, X., Das, K. C.: Some results on the Laplacian spread of a graph.Linear Algebra Appl. 505 (2016), 245-260. Zbl 1338.05158, MR 3506494, 10.1016/j.laa.2016.05.002 |
| Reference:
|
[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. Zbl 1240.05186, MR 2810546, 10.1007/s11766-011-2732-4 |
| Reference:
|
[7] Chung, F. R. K.: Spectral Graph Theory.Regional Conference Series in Mathematics 92. AMS, Providence (1997). Zbl 0867.05046, MR 1421568, 10.1090/cbms/092 |
| Reference:
|
[8] Edwards, C. S.: The largest vertex degree sum for a triangle in a graph.Bull. Lond. Math. Soc. 9 (1977), 203-208. Zbl 0357.05058, MR 0463005, 10.1112/blms/9.2.203 |
| Reference:
|
[9] Fath-Tabar, G. H., Ashrafi, A. R.: Some remarks on Laplacian eigenvalues and Laplacian energy of graphs.Math. Commun. 15 (2010), 443-451. Zbl 1206.05062, MR 2814304 |
| Reference:
|
[10] Goldberg, F.: Bounding the gap between extremal Laplacian eigenvalues of graphs.Linear Algebra Appl. 416 (2006), 68-74. Zbl 1107.05059, MR 2232920, 10.1016/j.laa.2005.07.007 |
| Reference:
|
[11] Grone, R., Merris, R.: The Laplacian spectrum of graph. II.SIAM J. Discrete Math. 7 (1994), 221-229. Zbl 0795.05092, MR 1271994, 10.1137/S0895480191222653 |
| Reference:
|
[12] Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total $\phi $-electron energy of alternant hydrocarbons.Chem. Phys. Lett. 17 (1972), 535-538. 10.1016/0009-2614(72)85099-1 |
| Reference:
|
[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. Zbl 1312.05082, MR 3379554, 10.3103/S106836231405001X |
| Reference:
|
[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. Zbl 1326.05082, MR 3338961, 10.1017/S0004972715000027 |
| Reference:
|
[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. MR 1555027, 10.1007/BF02418571 |
| Reference:
|
[16] Kemeny, J. G., Snell, J. L.: Finite Markov Chains.The University Series in Undergraduate Mathematics. Van Nostrand, Princeton (1960). Zbl 0089.13704, MR 0115196 |
| Reference:
|
[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. Zbl 1426.05101, MR 3743658, 10.1016/j.amc.2017.12.003 |
| Reference:
|
[18] Merris, R.: Laplacian matrices of graphs: A survey.Linear Algebra Appl. 197-198 (1994), 143-176. Zbl 0802.05053, MR 1275613, 10.1016/0024-3795(94)90486-3 |
| Reference:
|
[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. 10.5937/SPSUNP1501025M |
| Reference:
|
[20] Milovanović, I. Ž., Milovanović, E. I., Glogić, E.: On Laplacian eigenvalues of connected graphs.Czech. Math. J. 65 (2015), 529-535. Zbl 1363.15016, MR 3360442, 10.1007/s10587-015-0191-4 |
| Reference:
|
[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 |
| Reference:
|
[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). Zbl 0771.26009, MR 1220224, 10.1007/978-94-017-1043-5 |
| Reference:
|
[23] Nikiforov, V.: The energy of graphs and matrices.J. Math. Anal. Appl. 326 (2007), 1472-1475. Zbl 1113.15016, MR 2280998, 10.1016/j.jmaa.2006.03.072 |
| Reference:
|
[24] Nordhaus, E. A., Gaddum, J. W.: On complementary graphs.Am. Math. Mon. 63 (1956), 175-177. Zbl 0070.18503, MR 0078685, 10.2307/2306658 |
| Reference:
|
[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 |
| Reference:
|
[26] Palacios, J. L.: Some inequalities for Laplacian descriptors via majorization.MATCH Commun. Math. Comput. Chem. 77 (2017), 189-194. Zbl 1466.92279, MR 3645376 |
| Reference:
|
[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. 10.1002/qua.22396 |
| Reference:
|
[28] Shi, L.: Bounds on Randić indices.Discr. Math. 309 (2009), 5238-5241. Zbl 1179.05039, MR 2548924, 10.1016/j.disc.2009.03.036 |
| Reference:
|
[29] Shi, L., Wang, H.: The Laplacian incidence energy of graphs.Linear Algebra Appl. 439 (2013), 4056-4062. Zbl 1282.05152, MR 3133474, 10.1016/j.laa.2013.10.028 |
| Reference:
|
[30] You, Z., Liu, B.: On the Laplacian spectral ratio of connected graphs.Appl. Math. Lett. 25 (2012), 1245-1250. Zbl 1248.05116, MR 2947387, 10.1016/j.aml.2011.09.071 |
| Reference:
|
[31] You, Z., Liu, B.: The Laplacian spread of graphs.Czech. Math. J. 62 (2012), 155-168. Zbl 1245.05089, MR 2899742, 10.1007/s10587-012-0003-z |
| Reference:
|
[32] Zhou, B.: On sum of powers of the Laplacian eigenvalues of graphs.Linear Algebra Appl. 429 (2008), 2239-2246. Zbl 1144.05325, MR 2446656, 10.1016/j.laa.2008.06.023 |
| Reference:
|
[33] Zumstein, P.: Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph: Diploma Thesis.ETH, Zürich (2005). |
| . |
