Signless Laplacian Extremal Hypergraphs

• The paper establishes that signless Laplacian spectral extremal hypergraphs maximize the spectral radius under Turán-type forbidden subhypergraph constraints.
• It leverages tensor eigenvalue techniques and reduction criteria, using Hölder and Young inequalities to derive precise spectral bounds.
• Practical characterizations for supertrees, power hypergraphs, and the Fano plane case demonstrate actionable structural optimizations.

A signless Laplacian spectral extremal hypergraph is an $r$-uniform hypergraph that maximizes the spectral radius of its signless Laplacian tensor or matrix, often under combinatorial or structural constraints such as forbidden subhypergraphs or prescribed invariants. The spectral properties of the signless Laplacian capture and generalize important extremal features from classical graph theory — notably, Turán-type questions — into the regime of uniform hypergraphs, incorporating higher-order structures and tensorial eigenvalue problems. Recent work establishes precise extremal results and reduction criteria for the signless Laplacian spectral radius in the context of forbidden subhypergraphs and for special classes such as supertrees and power hypergraphs.

1. Fundamental Definitions and Framework

An $r$-uniform hypergraph $\mathcal{H}=(V,E)$ comprises a vertex set $V$ of size $n$ and a hyperedge set $E\subseteq \binom{V}{r}$. The adjacency tensor $\mathcal{A}(\mathcal{H})$ is an order-$r$ dimension-$n$ symmetric tensor with

$$a_{i_1i_2\cdots i_r} = \begin{cases} \frac{1}{(r-1)!}, & \{v_{i_1},\dots,v_{i_r}\}\in E, \ 0, & \text{otherwise.} \end{cases}$$

The degree tensor $\mathcal{D}(\mathcal{H})$ is diagonal with $d_{ii\cdots i} = d_\mathcal{H}(v_i)$, the degree of vertex $v_i$.

The signless Laplacian tensor is defined as $$\mathcal{Q}(\mathcal{H}) = \mathcal{D}(\mathcal{H}) + \mathcal{A}(\mathcal{H})$$. Its eigenvalues are scalars $\lambda$ satisfying $\mathcal{Q}(\mathcal{H}) x = \lambda x^{[r-1]}$ for some non-zero $x\in\mathbb{C}^n$; the signless Laplacian spectral radius $q(\mathcal{H}) = \rho(\mathcal{Q}(\mathcal{H}))$ denotes the eigenvalue of largest modulus (Lu et al., 13 Jan 2026, Duan et al., 2018).

In alternative matrix-based approaches — especially for $k$-uniform hypergraphs — the signless Laplacian matrix $Q(H) = BB^\top$, where $B$ is the vertex-edge incidence matrix. The entries of $Q(H)$ are $Q_{v,v} = d(v)$ and $Q_{v,w}$ counts the number of hyperedges containing both $v$ and $w$. This matrix is real symmetric, non-negative, and positive semidefinite; its spectral radius is denoted $\rho(H)$ (Cardoso et al., 2019).

2. The Signless Laplacian Spectral Turán Problem

The signless Laplacian spectral Turán problem generalizes classical Turán extremal questions: Given a family $\mathcal{F}$ of forbidden $r$-graphs and integer $n$, determine

$$q_{\max}(n, \mathcal{F}) := \max\{ q(G) : G \text{ is } \mathcal{F}\text{-free } r\text{-graph on } n \text{ vertices} \},$$

and characterize the extremal $r$-graphs achieving this maximum.

A central recent development is a reduction criterion: If $\mathcal{F}$ is degree-stable with respect to a family $\mathcal{H}_n$ (meaning that $\mathcal{F}$-free $G$ with high minimum degree must lie in $\mathcal{H}_n$), and if the following “natural assumptions” hold:

• (i) The increment $\ex_r(n,\mathcal{F})-\ex_r(n-1,\mathcal{F})$ grows as $\pi(\mathcal{F})/(r-1)!\, n^{r-1} + O(n^{r-1})$ for $\pi(\mathcal{F}) > 1/2$.
• (ii) $q(\mathcal{H}_n)$ is approximated by $2r\,\ex_r(n,\mathcal{F})/n + O(n^{r-2})$, then for $n\gg 1$,

$q(G) \le \max_{H\in\mathcal{H}_n} q(H),$

with equality only if $G\in\mathcal{H}_n$ (Lu et al., 13 Jan 2026).

The proof leverages the tensor eigenequations and employs inequalities of Hölder and Young to force high minimum degree, followed by a vertex-removal induction based on the principal eigenvector.

3. Extremal Results for Supertrees and Structural Constraints

For $k$-uniform supertrees (connected, acyclic $k$-uniform hypergraphs), extremal constructions of maximum signless Laplacian spectral radius are obtained for classes characterized by given diameter or pendent structure (Duan et al., 2018):

• Within the class $\mathbb{S}(m, d, k)$ of $k$-uniform supertrees with $m$ edges and diameter $d$, the unique maximizer is $S_1(m, d, k)$: a loose path of length $d$ with all remaining $m-d$ edges attached as pendent edges at the central path vertex.
• The same grafting (edge-releasing) techniques optimize structure in the classes $\mathbb{T}(n,p,k)$ (supertrees with $p$ pendent edges) and $\mathbb{G}(n,q,k)$ (supertrees with $q$ pendent vertices), showing that mass concentration at a central vertex or the unique BFS ordering yields extremality for $q(G)$.

The eigenpairs organize into characteristic polynomial systems whose largest root yields $q(S_1(m,d,k))$, uniquely determined by the structural symmetries and the Perron–Frobenius theorem for nonnegative symmetric tensors.

4. Spectral Inequalities and Structural Parameters

The signless Laplacian spectrum provides sharp bounds and characterizations involving hypergraph invariants (Cardoso et al., 2019):

• Degree bounds: For a $k$-uniform connected hypergraph $H$,

$$k {\cdot} d(H) \le \rho(H) \le k {\cdot} \Delta(H),$$

with equality characterizing regularity.

• Edge-degree bounds:

$$\min_{e\in E} \sum_{v\in e} d(v) \le \rho(H) \le \max_{e\in E} \sum_{v\in e} d(v).$$

• Chromatic number bound: For chromatic number $\chi(H)$,

$\chi(H) \le \frac{\rho(H)}{k} + 1,$

achieving equality for the complete $k$-graph $K_n^k$.

• Diameter and eigenvalue multiplicity: The number $t$ of distinct eigenvalues of $Q(H)$ satisfies $t \ge D + 1$ for diameter $D$; spectral gap bounds further constrain $D$ in terms of the largest eigenvalues and the principal eigenvector's minimal coordinate.

These results generalize classical graph-theoretic relationships, establishing that the spectral radius encodes both local (degree-based) and global (chromatic, diameter) extremal features.

5. Power Hypergraphs and Zero Eigenvalue Multiplicity

The spectrum of the signless Laplacian for power hypergraphs is determined directly from the base hypergraph: For the generalized power hypergraph $H_s^r$ constructed from a $k$-uniform $H$, the nonzero eigenvalues $\mu_i$ of $Q(H)$ yield eigenvalues $s(\mu_i-k)+r$ of $Q(H_s^r)$, along with additional eigenvalues $r-ks$ and $0$ of specified multiplicities. This explicit spectral decomposition enables transfer of extremality and tight bound results from the base to its powers.

A hypergraph is partially bipartite if $V=V_0\cup V_1\cup V_2$ with $V_1,V_2\neq\varnothing$ and every edge not contained in $V_0$ meets both $V_1$ and $V_2$. The presence and multiplicity of zero in the spectrum of $Q(H)$ exactly characterize such partial bipartiteness. For balanced ratios $|e\cap V_1|/|e\cap V_2|=c$ over all relevant $e$, $0$ is guaranteed as an eigenvalue (Cardoso et al., 2019).

6. Concrete Extremal Characterizations: The Fano Plane Case

The Fano plane $\mathrm{PG}_2(2)$ is the unique 3-uniform hypergraph on 7 points with edge-set $\{123,345,561,174,275,376,246\}$. For this case:

• Turán and stability theorems establish that extremal $\mathrm{PG}_2(2)$-free triple systems of maximum size are 2-colorable, and further that high minimum degree forces this structure.
• The extremal structures are the balanced complete bipartite triple systems $\mathcal{B}_n$ (2-colorable, balanced).
• The signless Laplacian spectral extremal hypergraph among all $\mathrm{PG}_2(2)$-free 3-graphs on $n \gg 1$ vertices is the balanced complete bipartite triple system, achieving

$$q(\mathcal{B}_n) = \begin{cases} \frac{3}{4} n^2 - \frac{3}{2} n, & n \text{ even}, \ \frac{3}{4} n^2 - \frac{3}{2} n - \frac{3}{4} + \frac{3}{2n}, & n \text{ odd}. \end{cases}$$

Equality holds only for $G = \mathcal{B}_n$. Thus, the signless Laplacian spectral Turán problem for the Fano plane is resolved by leveraging the general reduction criterion and degree-stability, reducing the spectral extremal problem to a computation on classical Turán-extremal constructions (Lu et al., 13 Jan 2026).

7. Perspectives and Implications

Recent advances establish that for broad classes of forbidden configurations and structural constraints, extremal signless Laplacian spectral properties are governed by high-degree stability and controlled growth properties (“natural assumptions”). The reduction theorems in (Lu et al., 13 Jan 2026) provide a blueprint for resolving spectral extremal problems in $r$-uniform hypergraphs, conditional on classical Turán-existence and stability arguments. Structural theorems for supertrees and power hypergraphs (Duan et al., 2018, Cardoso et al., 2019) further clarify the optimizing configurations under additional invariants.

A plausible implication is that future research may exploit these reductions and spectral–structural correspondences to resolve new hypergraph extremal problems and extend the framework to other spectral tensors, strengthening the interplay between combinatorial extremal theory and spectral analysis.