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arXiv · 2512.10364

Spectral bounds for vertex-weighted Laplacians of simplicial complexes

Yueli Han·Lu Lu
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Abstract

The vertex-weighted Laplacian naturally extends the combinatorial Laplacian for simplicial complexes. Inspired by Lew's foundational techniques for vertex-weighted Laplacians, we present a comprehensive spectral analysis of this operator. First, we determine how basic operations, including joins, complements, and Alexander duals, affect its spectrum. This yields a sharp upper bound on the spectral radius in terms of vertex weights, along with a lower bound on the multiplicity at which this bound is attained. Second, we establish a sharp lower bound for the spectral gap and characterize when the equality holds. Third, explicit lower bounds for the remaining eigenvalues are derived, linking the vertex-weighted Laplacian spectrum to that of a related weighted graph. Finally, we reveal new spectral relations between a simplicial complex and its subcomplexes. These results not only generalize numerous known theorems on combinatorial Laplacians but also provide deeper spectral insights into simplicial structures, ultimately unifying and extending a broad range of earlier work in this field.

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Yueli Han, Lu Lu. 2025-12-11. Spectral bounds for vertex-weighted Laplacians of simplicial complexes. https://arxiv.org/abs/2512.10364

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