Signless Laplacian Spectral Turán Problem

• The paper establishes the asymptotic bounds for the signless Laplacian spectral Turán function by linking graph structure with forbidden subgraphs.
• It employs Rayleigh quotient, vertex deletion, and stability techniques to analyze extremal graphs like balanced complete multipartite graphs and split graphs.
• The study extends classical results to hypergraphs and simplicial complexes, highlighting open challenges in the bipartite and multidimensional settings.

The Signless Laplacian Spectral Turán Problem seeks, for a given forbidden subgraph $F$ and integer $n$, the explicit determination or sharp asymptotics of

$$\mathrm{ex}_q(n,F) := \max\{ q(G) : |V(G)| = n,\ F \not\subseteq G \},$$

where $q(G)$ is the largest eigenvalue (the Q-index, or signless Laplacian spectral radius) of $Q(G) = D(G) + A(G)$, with $D(G)$ the degree matrix and $A(G)$ the adjacency matrix. This problem generalizes classical Turán extremal graph theory into a spectral context and reveals structural and spectral analogues for graph families, cycles, color-critical graphs, and more, with further extensions to hypergraphs and simplicial complexes.

1. Formulation and Basic Definitions

Let $G$ be a simple $n$-vertex graph. The signless Laplacian is defined as

$Q(G) = D(G) + A(G),$

where $D(G)$ is the diagonal matrix of vertex degrees and $A(G)$ the adjacency matrix. The spectral radius—maximum eigenvalue of $Q(G)$—is denoted $q(G)$. For a forbidden subgraph $F$, the signless Laplacian spectral Turán function is

$$\mathrm{ex}_q(n,F) = \max\{q(G)\colon |V(G)| = n,\ F \not\subseteq G\}.$$

The central question is to determine asymptotic or exact values for $\mathrm{ex}_q(n,F)$ and identify all extremal graphs attaining this maximum.

2. General Theorems and Chromatic Threshold

The analogue of the Erdős–Stone–Simonovits theorem—i.e., the spectral Turán theorem—has been developed for $q(G)$. For any $F$ with chromatic number $\chi(F) \geq 3$,

$$\mathrm{ex}_q(n, F) = \left( 1 - \frac{1}{\chi(F)-1} + o(1) \right) 2n,$$

and the limit

$$T_q(F) := \lim_{n \to \infty} \mathrm{ex}_q(n, F)/n = 2\left(1 - \frac{1}{\chi(F)-1}\right)$$

holds. The extremal examples are balanced complete $(\chi(F) - 1)$-partite graphs $T_{\chi(F)-1}(n)$. This result extends both the classical extremal formula for the number of edges and Nikiforov’s spectral version for the adjacency matrix. Precise asymptotics and an explicit $o(1)$-term (but not a named constant) are given for the error, and the case of bipartite $F$ (e.g., even cycles) is excluded due to known counterexamples (Zheng et al., 16 Feb 2025).

Class of $F$	Asymptotic $\mathrm{ex}_q(n, F)$	Extremal Structure
$\chi(F) \geq 3$	$(1 - 1/(\chi(F)-1) + o(1)) 2n$	Balanced complete $(\chi(F)-1)$-partite graph
$\chi(F) = 2$	Not of above form; exceptions for bipartite $F$	Problem open

3. Special Families and Exact Results

Research has provided exact and asymptotic answers for various forbidden subgraphs:

• Complete graphs and colors: Forbidding $K_{r+1}$ or any color-critical $F$ with $\chi(F) = r+1 \geq 3$, the unique extremal graph maximizing $q(G)$ is $T_{n,r}$, and $q(T_{n,r}) = 2n(1-1/r) + O(1)$ (Zheng et al., 10 Apr 2025, Wu et al., 1 Dec 2025).
• Odd and even cycles: For $C_{2k+1}$ or $C_{2k+2}$-free graphs, the extremal graph is $S_{n,k} = K_k \vee (n-k)K_1$ (a $k$-clique joined to an independent set). For $C_{2k+2}$, this structure optimizes $q(G)$ for large $n$ but does not conform to the Turán-type density formula, highlighting a key deviation for bipartite $F$ (Chen et al., 2021).
• Fans and intersecting triangles: For the $k$-fan (the graph $F_k=K_1 \vee kK_2$), the unique extremal is again $S_{n,k}$ for $n \geq 3k^2 - k - 2$, with

$$q(S_{n,k}) = \frac{ (n + 2k - 2) + \sqrt{ (n + 2k - 2)^2 - 8k(k-1) } }{2 }$$

(Zhao et al., 2020).

• Linear forests and theta graphs: For certain linear forests (forests with at most two odd paths) and forbidden theta graphs $\theta(l_1,l_2,l_3)$, the extremal graphs are split graphs $S_{n,h}$, $S_{n,k}$, their slight modifications, or friendship graphs depending on forbidden pattern (Liu et al., 2024, Chen et al., 2020).
• Trees and the Erdős–Sós context: Forbidding all trees of order $2k+2$ (or $2k+3$), the extremal graph is $S_{n,k}$ ($K_k \vee (n-k)K_1$) or its variant $S_{n,k}^+$, the latter adding an edge within the independent set, generalizing spectral analogues of classical tree extremal theory (Chen et al., 2022).

Forbidden Structure	Extremal Graph	Exact $q(G)$/Formula
$K_{r+1}$-free, $\chi(F) \ge 3$	$T_{n,r}$	$2n(1-1/r) + O(1)$
$C_{2k+1}$-free or $F_k$-free	$S_{n,k}$	Closed form in $n,k$
Tree of order $2k+2$ forbidden	$S_{n,k}$	$\frac{n+2k-2 + \sqrt{ (n+2k-2)^2 -8(k^2-k) }}{2}$
Trees of order $2k+3$ forbidden	$S_{n,k}^+$	Real root of associated cubic
$K_{s,t}$-minor free (large $n$)	$F_{s,t}(n)=K_{s-1}\vee(p\cdot K_t\cup K_r)$	Polynomial for $q(G)$

4. Proof Techniques and Methodology

Most proofs employ a combination of:

• Rayleigh quotient maximization and explicit Perron vector construction/analysis.
• Vertex deletion and eigenvector minimality (removing the smallest coordinate to perform induction or contradiction).
• Degree threshold/stability reduction: Forbidding $F$ or imposing a high $q(G)$ forces "almost" $r$-partite or split structure through degree sum/Cauchy interlacing arguments.
• Structure theorems from extremal combinatorics (e.g., complete bipartite subgraphs must contain all large trees).
• Equitable partition and quotient matrix reductions for computation of Q-index.
• Monotonicity and local perturbation arguments: Any deviation from the extremal construction decreases $q(G)$.
• Spectral stability theorems: If $q(G)$ is near the extremal value, $G$ is close in edit distance to an extremal configuration (Zheng et al., 10 Apr 2025, Chen et al., 2021, Zheng et al., 16 Feb 2025, Zheng et al., 3 Jul 2025).

5. Extensions: Hypergraphs and Simplicial Complexes

Recent advances extend the signless Laplacian spectral Turán problem to:

• $r$-uniform hypergraphs: Formulated with $r$-order tensors $$\mathcal{Q}(\mathcal{H}) = \mathcal{D}(\mathcal{H}) + \mathcal{A}(\mathcal{H})$$, where spectral extremality is obtained through reduction to “degree-stable” families. For example, for the Fano plane, the extremal $3$-graph is the balanced bipartite $3$-graph $\mathcal{B}_n$, with $q(\mathcal{B}_n)$ exactly computed (Lu et al., 13 Jan 2026).
• Simplicial complexes: The signless Laplacian is extended as $Q_{i}^{up}(K)$ for the $i$-th up Laplacian on $i$-faces of a complex $K$. The extremal spectral radius, for pure $r$-dimensional, $r$-hole-free complexes, is attained by the tented complex $T_n^r$, with

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

yielding bounds for both hypergraph and simplicial complex Turán numbers (Fan et al., 30 Jul 2025).

6. Open Problems and Current Directions

Key open questions and conjectures include:

• Sharp $(n,o(1))$ asymptotics and stability for ex$_q(n, F)$ beyond color-critical, non-bipartite cases.
• Bipartite forbidden subgraphs (e.g., even cycles $C_{2k}$, $K_{s,t}$): The classical Turán density formula fails; extremal value and structure remain open except for some cases such as $C_4$.
• Higher-dimensional and operator generalizations: For p-Laplacians, other normalized variants, and further combinatorial structures.
• Comparison and exact coincidence with edge-Turán extremals: Conjectures propose spectral and classical extremals always coincide for $F$ with $ex(n, F) = e(T_{n, r}) + O(1)$ for large $n$.
• Hypergraph and complex stability: Extension of “degree-stable” and spectral removal lemmas.

7. Survey and Synthesis

A synthesis of the known results demonstrates a dichotomy:

• For color-critical, non-bipartite $F$, the spectral Turán problem behaves identically to its classical extremal analogue, with Turán graphs as unique extremals.
• For bipartite $F$, linear forests, trees, and theta-graphs, the extremal structures become split graphs, stars, or other special constructions.

Research continues toward:

• Sharper understanding for bipartite $F$,
• Extensions to hypergraphs and complexes,
• Stability and uniqueness of spectral extremals,
• Identification of signless Laplacian extremals for complex or non-Zykov-symmetric forbidden families.

The field leverages deep interplay between combinatorial extremality, stability methods, and spectral theory, pushing the boundary of extremal and spectral combinatorics (Li et al., 2021, Zheng et al., 16 Feb 2025, Zheng et al., 10 Apr 2025, Fan et al., 30 Jul 2025, Lu et al., 13 Jan 2026).