Spectral Extremal Theory in Simplicial Complexes

• Spectral theory of simplicial complexes is an analytical framework linking eigenvalue bounds of the signless Laplacian to topological invariants and combinatorial structure.
• The tented complex serves as a model for hole-free complexes, maximizing the spectral radius by ensuring full (r–1)-skeleton and connectivity.
• Sharp spectral inequalities derived via Perron-Frobenius theory refine classical Turán numbers by connecting spectral extremality with Betti numbers.

The spectral extremal theory for simplicial complexes—particularly the paper of the signless Laplacian spectral radius in relation to the absence of holes—provides a unified framework linking algebraic, topological, and combinatorial extremal problems. Recent developments have characterized the extremal simplicial complexes maximizing the (up) signless Laplacian spectral radius under topological constraints, provided sharp spectral inequalities in terms of Betti numbers, and leveraged these results to refine upper bounds on classical Turán numbers for hypergraphs and complexes.

1. Spectral Extremal Characterization for Hole-Free Complexes

For a pure $r$-dimensional simplicial complex $K$ on $n$ vertices, the (up) signless Laplacian $Q_{r-1}^{\mathrm{up}}(K)$ acts on $(r-1)$-faces and encodes, via its spectral radius $q_{r-1}(K)$, intricate information about the complex’s higher-order combinatorial connectivity. Among all hole-free complexes—specifically, complexes with trivial $i$-th homology for $1 \leq i \leq r$, equivalently without any subcomplex homeomorphic to the $r$-sphere—the maximizer of $q_{r-1}(K)$ must satisfy two key structural properties:

• It contains the full $(r-1)$-skeleton (i.e., every $(r-1)$-subset of the vertex set is a face).
• It is $(r-1)$-path connected; that is, its (bipartite) incidence graph $B_{r-1}(K)$ is connected.

Any complex lacking these properties may be strictly improved (in $q_{r-1}$) by adding missing faces or joining disconnected parts without introducing new holes, as formalized in Lemma 2.3 and its corollaries.

The principal example—referred to in the literature as the "tented complex" $T_n^r$—is the pure $r$-dimensional complex whose facets are precisely the $(r+1)$-sets containing a distinguished vertex (say, vertex $n$):

$$T_n^r = \left\{ \{n\} \cup F : F \in \binom{[n-1]}{r} \right\}.$$

This construction ensures $(r-1)$-connectivity and full lower skeleton, frequently realizing the extremal spectral radius.

In addition, for odd $r \geq 3$, complexes containing certain "rhombic" subcomplexes (subdivisions homeomorphic to $S^r$) may match the extremal value, although these represent exceptions rather than the typical structure (Theorem 3.6).

2. Sharp Upper Bounds on the Signless Laplacian Spectral Radius

Given the above structural considerations, the spectral radius $q_{r-1}(K)$ is subject to a uniform upper bound. For any pure $r$-dimensional simplicial complex $K$ with trivial $i$th homology for $1 \leq i \leq r$ (i.e., no holes of positive dimension), the following inequality holds:

$q_{r-1}(K) \leq r n - r^2 + 1,$

where $n = |V(K)|$ is the number of vertices. Equality is achieved precisely by the tented complex $T_n^r$ for even $r$, and either $T_n^r$ or a complex containing a rhombic subcomplex for odd $r \geq 3$.

For complexes with prescribed $r$-th Betti number $\beta_r(K) = t > 0$, the bound generalizes to

$q_{r-1}(K) \leq r n - r^2 + t + 1,$

showing explicitly how topological complexity (number of $r$-dimensional "holes") forces possible spectral extremality upward.

These bounds are proven using Perron-Frobenius theory and detailed analysis of the signless Laplacian’s action, leveraging the maximality properties of the all-ones eigenvector and the required connectivity of the supporting incidence graph (Theorem 3.6, Theorem 3.10).

3. Connection to Turán-Type Extremal Problems

The signless Laplacian spectral radius intimately constrains the face numbers in a simplicial complex, thereby linking spectral and classical extremal (Turán-type) combinatorics. The number of facets $|S_{r+1}(K)|$ can be upper bounded in terms of $q_{r-1}(K)$ via

$$|S_{r+1}(K)| \leq \frac{q_{r-1}(K) \cdot \binom{n}{r+1}}{(r+2)^2}.$$

Substituting the spectral extremal values produces explicit upper bounds on Turán numbers—e.g., for the avoidance of specific forbidden complexes homeomorphic to spheres $S^r$ (Corollary 3.8). For instance, for $r=3$,

$$\text{ex}(n, \Delta_4^{3}) \leq \frac{1}{9} n^3 - o(n^3),$$

where $\text{ex}(n, \Delta_4^{3})$ is the maximum number of facets in an $n$-vertex, 3-dimensional complex avoiding the $4$-vertex sphere.

This approach recovers classical results for $r=1$ (bipartite graphs being extremal for Turán’s theorem) and yields new sharp asymptotics for higher-order complexes, situating spectral tools as competitive and sometimes superior to traditional combinatorial methods for extremal problems on hypergraphs and complexes.

4. Topological and Algebraic Implications

These results reveal that the signless Laplacian spectral radius is a powerful invariant, encoding both topological (homological) and combinatorial data. Absence of holes, as detected via Betti numbers, is essential for maximizing $q_{r-1}$; otherwise, any increase in topological complexity (e.g., more $r$-parametric cycles) translates directly to a strictly lower possible spectral radius, barring the exceptional rhombic subcomplexes (for odd $r$).

In particular, the unique spectral extremality of the tented complex (for even $r$) and its generalization to more complex Betti structures suggests a strong form of rigidity: the spectrum can often "detect" the presence or absence of holes, and the structural combinatorial properties required for maximizing the spectral radius are forced by these topological constraints.

5. Directions for Further Study

Several open issues remain, particularly for odd $r \geq 3$, where the presence of exceptional equality cases involving rhombic or spherical subcomplexes complicates full classification. Understanding the precise structure and abundance (or rarity) of such extremal complexes is a direction of ongoing research.

Potential further extensions include:

• Extending the spectral extremal framework to families of complexes with controlled lower-dimensional homology or partially "filled" skeletons;
• Refining asymptotic estimates for Turán-type numbers using sharper spectral-combinatorial inequalities;
• Investigating the higher-order spectral invariants of other Laplacian variants (combinatorial, normalized, or reduced) within topologically constrained classes.

Additionally, exploring applications in data science (e.g., topological data analysis, spectral clustering of higher-order structures) and theoretical computer science (property testing, hard instances for expansion) may leverage these spectral extremal results.

Summary Table

Property	Extremal Complex (no holes)	Upper Bound
Full (r–1)-skeleton, (r–1)-connectivity	Tented complex $T_n^r$	$q_{r-1}(K) \le r n - r^2 + 1$
With Betti number $\beta_r(K) = t$	Tented $+$ $t$ holes	$q_{r-1}(K) \le r n - r^2 + t + 1$
Turán number (no $S^r$ subcomplex)	---	$$\text{ex}(n, \Delta_{r+1}^r) \le \left(\frac{(r-1)(n-r+1)+1}{r(n-r)}\right) \binom{n}{r}$$

These advances clarify the landscape for spectral extremal problems in high-dimensional combinatorics, elucidating the relationship between Laplacian spectra, topology, and extremal counting, and providing critical tools for further theoretical development (Fan et al., 30 Jul 2025).