Abstract
We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, \(N/(N-1)\leq \lambda_N\leq 2\), to the case of chemical hypergraphs.
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Notes
As for the case of graphs, it is clear by the definition of the normalized Laplacian that the spectrum of a hypergraph is given by the union of the spectra of its connected components. Therefore, without loss of generality we can choose to work on connected hypergraphs.
In the case of graphs, the multiplicity of \(0\) for the normalized Laplacian equals the number of connected components of the graph. Therefore, for connected graphs, \(\lambda_2\) is the first nonzero eigenvalue. The same doesn’t hold for hypergraphs, see [1]. This is why it is not yet clear how to generalize the Cheeger constant to chemical hypergraphs.
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Mulas, R. Sharp Bounds for the Largest Eigenvalue. Math Notes 109, 102–109 (2021). https://doi.org/10.1134/S0001434621010120
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DOI: https://doi.org/10.1134/S0001434621010120