Article
| 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: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 3 |
| Year: | 2022 |
| Pages: | 783-799 |
| Summary lang: | English |
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| Category: | math |
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| 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 |
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| Date available: | 2022-08-22T08:23:07Z |
| Last updated: | 2022-12-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150617 |
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