Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels

Abstract: An $r$-dimensional wheel is defined as the join of an $(r-2)$-simplex and a cycle. In this paper, we study the maximum signless Laplacian spectral radius of $n$-vertex $r$-dimensional pure simplicial complexes that contain no $r$-dimensional wheels. For sufficiently large $n$, we determine the extremal complexes that attain this maximum. Our result generalizes the corresponding extremal results of signless Laplacian on graphs and provides a spectral anlogue of a theorem of Sós, Erdős and Brown on the maximum number of facets of simplicial complexes in the case $r=2$.

• The paper establishes that the maximal signless Laplacian spectral radius in r-dimensional simplicial complexes without r-dimensional wheels is n.
• It employs high-dimensional Laplacian methods, combinatorial neighbor uniformity, and a Turán-type framework to derive sharp spectral bounds.
• It identifies the r-dimensional book as the unique extremal case under r-path connectivity, linking spectral properties to combinatorial rigidity.

Signless Laplacian Spectral Radius of Simplicial Complexes Without $r$-Dimensional Wheels

Introduction

The paper "Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels" (2604.04536) addresses the spectral Turán problem in the context of higher-dimensional combinatorial topology. Specifically, it investigates the extremal value of the signless Laplacian spectral radius among all pure $r$-dimensional simplicial complexes on $n$ vertices that avoid $r$-dimensional wheels as subcomplexes. The work generalizes fundamental results from spectral extremal graph theory, extending them to the domain of high-dimensional complexes, and provides spectral analogues of classical Turán-type results for simplicial complexes.

Background and Motivation

Extremal combinatorics and spectral theory have long been intertwined, with Turán-type extremal questions forming the backbone of the study of forbidden substructures. The classical Turán number $\textrm{ex}(n,F)$ counts the maximal number of edges in an $n$-vertex graph avoiding $F$. Spectral versions ask for the maximum spectral radius among such forbidden-subgraph graphs. For certain families—such as the complete graph, cycles, and wheels—these problems exhibit rich combinatorial and spectral structure. The generalization to $r$-uniform hypergraphs, and further to simplicial complexes, encounters substantial complexity due to the combinatorial intricacy of higher-dimensional adjacency and incidence structures.

Simplicial complexes and their associated Laplacians are now fundamental in topological combinatorics and data analysis. The signless Laplacian operator, in particular, encodes high-dimensional up-degree and adjacency relations. Studying extremal spectra of such operators under forbidden subcomplex constraints is pivotal for understanding structural, homological, and expansion properties of complexes.

Main Results

The core result establishes that for any pure $r$-dimensional simplicial complex $r$0 on $r$1 vertices avoiding $r$2-dimensional wheels (i.e., all subcomplexes of the form $r$3 for a cycle $r$4), the largest eigenvalue of the up signless Laplacian $r$5 is at most $r$6: $r$7 If, in addition, $r$8 is $r$9-path connected, then equality only holds for the $r$0-dimensional book $r$1—that is, the join of an $r$2-simplex and $r$3 isolated vertices.

This determination is sharp: the book complex achieves $r$4. These results synthesize and extend classical spectral extremal bounds for graphs (notably results for wheel-free graphs by Zhao et al. and for graphs without cycles), and they generalize a theorem of Sós, Erdős, and Brown related to the maximum number of facets in a wheel-free simplicial complex.

Methods

The authors leverage combinatorial, algebraic, and topological techniques:

• High-dimensional Laplacian Formalism: The $r$5-th up signless Laplacian captures adjacency between $r$6-faces and their $r$7-face extensions. The spectral radius is related to uniformity in the complex's down-neighbor structure.
• Turán Framework: By reducing the extremal question to counting $r$8-faces and their neighbor expansions under a forbidden subcomplex, the problem is handled using combinatorial and path connectivity arguments.
• Neighbor Uniformity: The concept of "1-neighbor uniformity" is essential: a complex is 1-neighbor uniform if each $r$9-face and each outside vertex determines exactly one down neighbor. This property is necessary and sufficient (under connectivity) for attaining spectral equality.
• Analysis of Cocycle Complexes: Structures generalizing tight cycles are precisely characterized using cocycle complexes, with intricate analysis on how wheel-like subcomplexes can or cannot arise.
• Use of Perron-Frobenius: The leading eigenvector is shown, in the extremal case, to be the all-ones vector, pinned down by equality cases in the uniform neighbor argument.

Illustrations provided clarify the nature of the forbidden subcomplexes and the structural transition from cycles to wheels in higher dimensions.

Figure 1: A triangulation of the M\"obius strip, illustrating the combinatorial structure of a tight cycle in dimension $n$0, which underpins the cocycle complex construction.

Figure 2: An illustration of the local combinatorial configuration in the proof of Lemma 3, specifically Case 2, depicting neighbor relations and obstruction to wheel formation.

Numerical and Structural Consequences

A standout assertion of the paper is that the extremal spectral value does not grow faster than linearly in $n$1 for large $n$2, regardless of $n$3: $n$4 This matches the exact bound for book complexes and is unattainable for other non-wheel-free or non-uniform complexes (subject to size constraints). Notably, for $n$5, this recovers the result that the extremal signless Laplacian spectral radius among $n$6-free graphs is achieved by the book graph $n$7.

Further, the characterization of equality cases for finite $n$8 reveals subtle combinatorial distinctions: for small $n$9, other types of cocycle complexes can arise, but for sufficiently large $r$0, only the $r$1-book is extremal.

Implications and Future Directions

The findings have several significant implications:

• Combinatorial Rigidity: The 1-neighbor uniformity required for spectral extremality demonstrates a form of combinatorial rigidity in higher dimensions, paralleling the uniqueness of Turán graphs in the graph case.
• Topological Expansion: Since spectral gaps and radii in Laplacians relate to expansion properties, these results contribute to understanding the "hardness" of facilitating cycles and wheels in high-dimensional expanders.
• Homological Bounds: The absence of wheels restricts the homology of the complex; spectral extremal results, therefore, interface with Betti number bounds and higher-order random walks.
• Generalization of Turán-Type Extremal Problems: These techniques are likely to be applicable to broader classes of forbidden subcomplexes beyond wheels, including higher genus surfaces and quasirandomness obstructions.

Further lines of research may address:

• The interplay between forbidden (hyper)graph minors and spectral extremal numbers in the complex setting.
• Extension to normalized Laplacian and signless Laplacian spectra.
• The effect of additional topological constraints, such as prescribed Betti numbers, on the spectral extremal value.
• Applications to random complexes and probabilistic topology, especially in the study of phase transitions for homological connectivity.

Conclusion

This work provides a thorough and precise characterization of the maximum signless Laplacian spectral radius in pure $r$2-dimensional simplicial complexes without $r$3-dimensional wheels, establishing the $r$4-dimensional book as the unique extremal configuration for large $r$5. The results bridge classical spectral extremal graph theory with higher-dimensional combinatorial topology and contribute foundational insights into the structure, expansion, and extremal spectra of high-dimensional complexes.