Inequalities for real number sequences with applications in spectral graph theory

Emina Milovanović; Şerife Burcu Bozkurt Altındağ; Marjan Matejić; Igor Milovanović

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 783-799
  • ISSN: 0011-4642

Abstract

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Let a = ( a 1 , a 2 , ... , a n ) be a nonincreasing sequence of positive real numbers. Denote by S = { 1 , 2 , ... , n } the index set and by J k = { I = { r 1 , r 2 , ... , r k } , 1 ≤ r 1 < r 2 < ⋯ < r k ≤ n } the set of all subsets of S of cardinality k , 1 ≤ k ≤ n - 1 . In addition, denote by a I = a r 1 + a r 2 + ⋯ + a r k , 1 ≤ k ≤ n - 1 , 1 ≤ r 1 < r 2 < ⋯ < r k ≤ n , the sum of k arbitrary elements of sequence a , where a I 1 = a 1 + a 2 + ⋯ + a k and a I n = a n - k + 1 + a n - k + 2 + ⋯ + a n . We consider bounds of the quantities R S k ( a ) = a I 1 / a I n , L S k ( a ) = a I 1 - a I n and S k , α ( a ) = ∑ I ∈ J k a I α in terms of A = ∑ i = 1 n a i and B = ∑ i = 1 n a i 2 . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

How to cite

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Milovanović, Emina, et al. "Inequalities for real number sequences with applications in spectral graph theory." Czechoslovak Mathematical Journal 72.3 (2022): 783-799. <http://eudml.org/doc/298414>.

@article{Milovanović2022,
abstract = {Let $a=(a_\{1\},a_\{2\},\ldots ,a_\{n\})$ be a nonincreasing sequence of positive real numbers. Denote by $S=\lbrace 1,2,\ldots ,n\rbrace $ the index set and by $J_\{k\}=\lbrace I= \lbrace r_\{1\},r_\{2\},\ldots ,r_\{k\} \rbrace $, $1\le r_\{1\}<r_\{2\}< \cdots <r_\{k\}\le n\rbrace $ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_\{I\}=a_\{r_\{1\}\}+a_\{r_\{2\}\}+\cdots +a_\{r_\{k\}\}$, $1\le k\le n-1$, $1\le r_\{1\}<r_\{2\}<\cdots <r_\{k\}\le n$, 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.},
author = {Milovanović, Emina, Bozkurt Altındağ, Şerife Burcu, Matejić, Marjan, Milovanović, Igor},
journal = {Czechoslovak Mathematical Journal},
keywords = {inequality; real number sequence; Laplacian eigenvalue of graph; normalized Laplacian eigenvalue},
language = {eng},
number = {3},
pages = {783-799},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inequalities for real number sequences with applications in spectral graph theory},
url = {http://eudml.org/doc/298414},
volume = {72},
year = {2022},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 783
EP - 799
AB - Let $a=(a_{1},a_{2},\ldots ,a_{n})$ be a nonincreasing sequence of positive real numbers. Denote by $S=\lbrace 1,2,\ldots ,n\rbrace $ the index set and by $J_{k}=\lbrace I= \lbrace r_{1},r_{2},\ldots ,r_{k} \rbrace $, $1\le r_{1}<r_{2}< \cdots <r_{k}\le n\rbrace $ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_{I}=a_{r_{1}}+a_{r_{2}}+\cdots +a_{r_{k}}$, $1\le k\le n-1$, $1\le r_{1}<r_{2}<\cdots <r_{k}\le n$, 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.
LA - eng
KW - inequality; real number sequence; Laplacian eigenvalue of graph; normalized Laplacian eigenvalue
UR - http://eudml.org/doc/298414
ER -

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