---
title: Signless Laplacian Spectral Turán Problem
url: https://www.emergentmind.com/topics/signless-laplacian-spectral-turan-problem
type: topic
---

# Signless Laplacian Spectral Turán Problem

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 [2502.11170].

| 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 $q(T_{n,r}) = 2n(1-1/r) + O(1)$ [2504.07852, 2512.01673].

- **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 $F$ [2108.03895].

- **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
\[
q(S_{n,k}) = \frac{ (n + 2k - 2) + \sqrt{ (n + 2k - 2)^2 - 8k(k-1) } }{2 }
\]
[2009.04738].

- **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 [2412.08417, 2007.08627].

- **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 [2209.03120].

| 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 [2504.07852, 2108.03895, 2502.11170, 2507.02263].

## 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 [2601.08595].

- **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 [2507.22518].

## 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 [2111.03309, 2502.11170, 2504.07852, 2507.22518, 2601.08595].

Source: https://www.emergentmind.com/topics/signless-laplacian-spectral-turan-problem