Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes

Abstract: Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\operatorname{up}}_{r-1}(K)$. It is known that [ λ{\max}\bigl(L^{\operatorname{up}}{r-1}(K)\bigr)\le n. ] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\widetilde H_{r-1}(K^c,\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected. Secondly, we prove a sharper upper bound for $λ{\max}(L^{\operatorname{up}}{r-1}(K))$: [ λ{\max}(L^{\operatorname{up}}{r-1}(K)) \le \max_{F\in S_r(K)} \bigl|\bigcup_{E \in \partial F} N_K(E) \bigr| \le n,] where, for an $(r-1)$-face $E$, $N_K(E)$ denotes the set of vertices $u$ outside $E$ such that the union $E \cup {u}$ is an $r$-face of $K$. This is the high-dimensional analog of the graph Laplacian bound. We give an explicit characterization of the equality case, and construct a broad family attaining the bound, namely, the partite semiregular complexes with admissible additions.

• The paper derives a universal upper bound for the largest Laplacian eigenvalue of simplicial complexes, showing it equals the number of vertices when homological conditions are met.
• It introduces a sharper locality-aware estimate based on neighborhood overlaps, generalizing Das' bound from graphs to higher dimensions.
• The study characterizes extremal complexes, including partite semiregular families, and links spectral extremality to non-trivial topological features.

Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes

Introduction

The paper "Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes" (2606.21233) investigates spectral extremal properties of combinatorial Laplacians in the setting of finite simplicial complexes. The focus is on deriving and characterizing upper bounds for the largest eigenvalue of the $(r-1)$-up Laplacian, extending classical results from graph theory to higher-dimensional combinatorial objects. Building on foundational results for graphs, particularly those by Anderson-Morley, Das, and subsequent generalizations, the authors develop a homological criterion and introduce sharper, locality-aware bounds for simplicial complexes, with explicit structural characterization of equality cases.

Universal Upper Bound and Homological Characterization

The classical result for graphs states that, for a graph $G$ on $n$ vertices, $\lambda_{\max}(L(G)) \leq n$ with equality if and only if the complement graph $G^c$ is disconnected. This paper generalizes this principle to $r$-dimensional simplicial complexes $K$ and proves that:

$\lambda_{\max}\bigl(_{r-1}(K)\bigr) \leq n$

where $n = |V(K)|$. Crucially, the bound is tight if and only if the reduced homology $\widetilde{H}_{r-1}(K^{c}, R)$ of the $G$0-dimensional complement $G$1 is non-trivial. This establishes a direct relationship between spectral extremality and topological non-triviality: complexes attaining maximal Laplacian eigenvalue exhibit non-zero homology in their complement. For $G$2, this reduces precisely to the classical graph case.

Sharper Local Upper Bounds: High-Dimensional Das-Type Estimates

The authors advance beyond the universal bound by providing a Das-type upper bound, sensitive to the local face structure of the simplicial complex:

$G$3

where, for an $G$4-face $G$5, $G$6 is the set of vertices outside $G$7 forming an $G$8-face with $G$9. The bound explicitly accounts for neighborhood overlaps—unlike prior degree-sum bounds—which enhances tightness. For $n$0, this reduces to Das' graph Laplacian bound: for an edge $n$1, the bound is $n$2. The proof leverages decomposition of the down Laplacian into local adjacency matrices on $n$3-vertex induced subcomplexes and derives a quadratic form through summation over $n$4-faces and local overlaps.

The paper also provides an explicit equality characterization: complex $n$5 attains the upper bound if and only if there exists a nonzero vector supported on $n$6-faces with maximal local number and annihilated by a certain local mapping $n$7. This equality criterion generalizes the mechanism behind extremal graphs for Das' bound to arbitrary dimension.

Structural Extremal Families: Partite Semiregular Complexes

The authors construct a broad family of complexes attaining the sharp bound: $n$8-partite semiregular complexes with admissible additions. These complexes mirror the extremal bipartite graph constructions in Das' theory and provide a systematic framework for extremal high-dimensional objects.

A semiregular $n$9-partite complex consists of $\lambda_{\max}(L(G)) \leq n$0 vertex classes with each $\lambda_{\max}(L(G)) \leq n$1-face containing precisely one vertex from each class and each suitable $\lambda_{\max}(L(G)) \leq n$2-subset appearing in exactly $\lambda_{\max}(L(G)) \leq n$3 faces. The admissible addition scheme allows for adding faces in a controlled manner—ensuring the local number condition and preservation of extremality—generalizing graph-theoretic operations (addition of edges within vertex classes) to higher dimensions. The paper rigorously proves that all complexes in this family saturate the upper bound.

Additionally, the authors demonstrate that, in higher dimensions, the converse does not always hold: there exist complexes attaining the sharp bound that are not partite semiregular or not contained in the constructed family. This highlights new phenomena in high-dimensional spectral extremality not present in the graph case.

Implications and Future Directions

The research establishes a nuanced spectral-topological correspondence, connecting Laplacian spectral extremality with homological properties, and refines spectral upper bounds by leveraging local face structures. The explicit classification of extremal complexes and generalization of graph-theoretic principles to higher dimensions advances the mathematical understanding of combinatorial Laplacians and their spectral behavior. The results may influence future developments in spectral combinatorics, topological data analysis, and discrete Hodge theory.

Practically, the sharp bounds and extremal families provide tools for analyzing, constructing, or certifying complexes with desired spectral properties—valuable in applications demanding tight control of Laplacian spectra, such as network synchronization, quantum computing, topological signal processing, and random walks on complexes. The explicit homological characterization offers means to detect structural or connectivity anomalies via spectral analysis.

Potential future research directions include extending the theory to weighted, directed, or non-pure complexes; investigating the behavior of eigenvalues beyond the spectral radius; elaborating on the converse problem in higher dimensions; and exploring algorithmic applications in computational topology, spectral clustering, and robustness analysis.

Conclusion

This paper advances the theory of combinatorial Laplacians on simplicial complexes by providing a universal upper bound linked to homological criteria and a sharper, locality-aware bound generalizing Das' extremal graph theory. Explicit structural classification of extremal complexes is achieved, with recognition of limitations and new phenomena in higher dimensions. The results deepen the spectral-topological landscape and set the stage for further theoretical and applied developments in high-dimensional spectral combinatorics.