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Article

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
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.
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