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Inequalities for real number sequences with applications in spectral graph theory. (English) Zbl 07584102

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.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
05C30 Enumeration in graph theory
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