Geodesic
Inequalities for real number sequences with applications in spectral graph theory
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 783-799
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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}
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 $J_{k}=\{I= \{ r_{1},r_{2},\ldots ,r_{k} \}$, $1\leq r_{1}$ the set of all subsets of $S$ of cardinality $k$, $1\leq k\leq n-1$. In addition, denote by $a_{I}=a_{r_{1}}+a_{r_{2}}+\cdots +a_{r_{k}}$, $1\leq k\leq n-1$, $1\leq r_{1}$, 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.
DOI : 10.21136/CMJ.2022.0155-21
Classification : 05C30, 15A18
Keywords: inequality; real number sequence; Laplacian eigenvalue of graph; normalized Laplacian eigenvalue 
@article{10_21136_CMJ_2022_0155_21,
     author = {Milovanovi\'c, Emina and Bozkurt Alt{\i}nda\u{g}, \c{S}erife Burcu and Mateji\'c, Marjan and Milovanovi\'c, Igor},
     title = {Inequalities for real number sequences with applications in spectral graph theory},
     journal = {Czechoslovak Mathematical Journal},
     pages = {783--799},
     year = {2022},
     volume = {72},
     number = {3},
     doi = {10.21136/CMJ.2022.0155-21},
     mrnumber = {4467942},
     zbl = {07584102},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0155-21/}
}
TY  - JOUR
AU  - Milovanović, Emina
AU  - Bozkurt Altındağ, Şerife Burcu
AU  - Matejić, Marjan
AU  - Milovanović, Igor
TI  - Inequalities for real number sequences with applications in spectral graph theory
JO  - Czechoslovak Mathematical Journal
PY  - 2022
SP  - 783
EP  - 799
VL  - 72
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0155-21/
DO  - 10.21136/CMJ.2022.0155-21
LA  - en
ID  - 10_21136_CMJ_2022_0155_21
ER  - 
%0 Journal Article
%A Milovanović, Emina
%A Bozkurt Altındağ, Şerife Burcu
%A Matejić, Marjan
%A Milovanović, Igor
%T Inequalities for real number sequences with applications in spectral graph theory
%J Czechoslovak Mathematical Journal
%D 2022
%P 783-799
%V 72
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0155-21/
%R 10.21136/CMJ.2022.0155-21
%G en
%F 10_21136_CMJ_2022_0155_21
Milovanović, Emina; Bozkurt Altındağ, Şerife Burcu; Matejić, Marjan; Milovanović, Igor. Inequalities for real number sequences with applications in spectral graph theory. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 783-799. doi: 10.21136/CMJ.2022.0155-21

[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 | MR | JFM

[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 | MR | JFM

[3] Butler, S. K.: Eigenvalues and Structures of Graphs: Ph.D. Thesis. University of California, San Diego (2008). | MR

[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 | MR | JFM

[5] Chen, X., Das, K. C.: Some results on the Laplacian spread of a graph. Linear Algebra Appl. 505 (2016), 245-260. | DOI | MR | JFM

[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 | MR | JFM

[7] Chung, F. R. K.: Spectral Graph Theory. Regional Conference Series in Mathematics 92. AMS, Providence (1997). | DOI | MR | JFM

[8] Edwards, C. S.: The largest vertex degree sum for a triangle in a graph. Bull. Lond. Math. Soc. 9 (1977), 203-208. | DOI | MR | JFM

[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 | JFM

[10] Goldberg, F.: Bounding the gap between extremal Laplacian eigenvalues of graphs. Linear Algebra Appl. 416 (2006), 68-74. | DOI | MR | JFM

[11] Grone, R., Merris, R.: The Laplacian spectrum of graph. II. SIAM J. Discrete Math. 7 (1994), 221-229. | DOI | MR | JFM

[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

[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 | MR | JFM

[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 | MR | JFM

[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 | MR

[16] Kemeny, J. G., Snell, J. L.: Finite Markov Chains. The University Series in Undergraduate Mathematics. Van Nostrand, Princeton (1960). | MR | JFM

[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 | MR | JFM

[18] Merris, R.: Laplacian matrices of graphs: A survey. Linear Algebra Appl. 197-198 (1994), 143-176. | DOI | MR | JFM

[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

[20] Milovanović, I. Ž., Milovanović, E. I., Glogić, E.: On Laplacian eigenvalues of connected graphs. Czech. Math. J. 65 (2015), 529-535. | DOI | MR | JFM

[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

[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 | MR | JFM

[23] Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326 (2007), 1472-1475. | DOI | MR | JFM

[24] Nordhaus, E. A., Gaddum, J. W.: On complementary graphs. Am. Math. Mon. 63 (1956), 175-177. | DOI | MR | JFM

[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

[26] Palacios, J. L.: Some inequalities for Laplacian descriptors via majorization. MATCH Commun. Math. Comput. Chem. 77 (2017), 189-194. | MR | JFM

[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

[28] Shi, L.: Bounds on Randić indices. Discr. Math. 309 (2009), 5238-5241. | DOI | MR | JFM

[29] Shi, L., Wang, H.: The Laplacian incidence energy of graphs. Linear Algebra Appl. 439 (2013), 4056-4062. | DOI | MR | JFM

[30] You, Z., Liu, B.: On the Laplacian spectral ratio of connected graphs. Appl. Math. Lett. 25 (2012), 1245-1250. | DOI | MR | JFM

[31] You, Z., Liu, B.: The Laplacian spread of graphs. Czech. Math. J. 62 (2012), 155-168. | DOI | MR | JFM

[32] Zhou, B.: On sum of powers of the Laplacian eigenvalues of graphs. Linear Algebra Appl. 429 (2008), 2239-2246. | DOI | MR | JFM

[33] Zumstein, P.: Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph: Diploma Thesis. ETH, Zürich (2005).

Cité par Sources :