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Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Essential Self-Adjointness

Marwa Ennaceur, Amel Jadlaoui

TL;DR

The paper develops geometry-free operator bounds and explicit criteria guaranteeing essential self-adjointness for discrete Hodge Laplacians on weighted simplicial complexes, avoiding reliance on geometric completeness or curvature. It introduces an adjacency–potential decomposition, uses Schur tests, and employs a line-complex reduction along with a unitary intertwiner between skew and symmetric coboundaries on colorable complexes to obtain sharp norm bounds and ESA. For the edge block on unweighted d-regular graphs, it proves the universal bound ||̃Δ_{1,*}|| ≤ 4(d−1), with weighted extensions via a comparability constant, and extends ESA results to higher degrees under finite dual up/down degrees. In periodic lattices, Floquet–Bloch analysis yields exact constants of order 2d for the edge block, illustrating both the sharpness gap and the efficacy of translation-invariant techniques. Overall, the work advances a robust operator-theoretic framework for discrete Hodge Laplacians, providing concrete, geometry-free tools with broad applicability to graphs, complexes, and lattice systems.

Abstract

We establish explicit operator norm bounds and essential self-adjointness criteria for discrete Hodge Laplacians on weighted graphs and simplicial complexes. For unweighted $d$-regular graphs we prove the universal estimate $\|\widetildeΔ_{1,*}\|\le 4(d-1)$, and we provide weighted extensions with a sharp comparability constant. These bounds apply without geometric completeness or curvature assumptions and ensure essential self-adjointness on natural cores. The approach extends to higher degrees via dual up/down degrees, and we show a unitary equivalence between skew and symmetric models on colorable complexes. For periodic lattices we complement the universal bounds with exact Floquet--Bloch constants, typically of order $2d$, illustrating both the sharpness in growth and the generality of our method.

Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Essential Self-Adjointness

TL;DR

The paper develops geometry-free operator bounds and explicit criteria guaranteeing essential self-adjointness for discrete Hodge Laplacians on weighted simplicial complexes, avoiding reliance on geometric completeness or curvature. It introduces an adjacency–potential decomposition, uses Schur tests, and employs a line-complex reduction along with a unitary intertwiner between skew and symmetric coboundaries on colorable complexes to obtain sharp norm bounds and ESA. For the edge block on unweighted d-regular graphs, it proves the universal bound ||̃Δ_{1,*}|| ≤ 4(d−1), with weighted extensions via a comparability constant, and extends ESA results to higher degrees under finite dual up/down degrees. In periodic lattices, Floquet–Bloch analysis yields exact constants of order 2d for the edge block, illustrating both the sharpness gap and the efficacy of translation-invariant techniques. Overall, the work advances a robust operator-theoretic framework for discrete Hodge Laplacians, providing concrete, geometry-free tools with broad applicability to graphs, complexes, and lattice systems.

Abstract

We establish explicit operator norm bounds and essential self-adjointness criteria for discrete Hodge Laplacians on weighted graphs and simplicial complexes. For unweighted -regular graphs we prove the universal estimate , and we provide weighted extensions with a sharp comparability constant. These bounds apply without geometric completeness or curvature assumptions and ensure essential self-adjointness on natural cores. The approach extends to higher degrees via dual up/down degrees, and we show a unitary equivalence between skew and symmetric models on colorable complexes. For periodic lattices we complement the universal bounds with exact Floquet--Bloch constants, typically of order , illustrating both the sharpness in growth and the generality of our method.
Paper Structure (34 sections, 15 theorems, 101 equations, 1 figure, 2 tables)

This paper contains 34 sections, 15 theorems, 101 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Combinatorial setting
  5. Graphs and simplicial complexes
  6. Hilbert cochains and Hodge operators
  7. Function spaces
  8. Cochain operators
  9. Laplacians via the Gauss--Bonnet operator
  10. Definition and block structure
  11. Local coordinate formulas
  12. Friedrichs form and first representation theorem
  13. Up-/down-Laplacians and Hodge identities
  14. Energy identities
  15. Normalization conventions and spectral footprint
  16. ...and 19 more sections

Key Result

Lemma 3.5

If $m_k(\sigma)>0$ for all $\sigma\in T_k$, then $\mathcal{C}^{k}_{c,*}(\mathcal{V})$ is dense in $\ell^2(m_k)$, $k=0,\dots,n$.

Figures (1)

  • Figure 1: Colorable triangle ($p=3$). Sorting by colors defines $S$; multiplication by $S$ yields the unitary $U$ intertwining skew and symmetric models.

Theorems & Definitions (45)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4: Weighted oriented $n$-simplicial complex
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Remark 4.1
  • Proposition 4.2
  • ...and 35 more