The signless Laplacian spectral Turán problems for hypergraphs

TL;DR

This work studies the signless Laplacian spectral radius $q(\mathcal{H})$ of $r$-uniform hypergraphs and develops a general degree-stability criterion that reduces signless Laplacian Turán problems to extremal constructions $\mathcal{H}_n$. When a family $\mathcal{F}$ is degree-stable with respect to $\mathcal{H}_n$ and the relevant ex$_r(n,\mathcal{F})$ and spectral approximations hold, the maximum $q$-value among $\mathcal{F}$-free hypergraphs on $n$ vertices is achieved by $\mathcal{H}_n$. As a concrete application, the authors fully characterize the signless Laplacian extremal hypergraph for the Fano plane, showing that for large $n$ the maximum is attained uniquely by the balanced 2-colorable 3-uniform hypergraph $\mathcal{B}_n$. These results extend the spectral Turán framework of Keevash, Lenz, and Mubayi to the signless Laplacian setting and suggest broad applicability to other degree-stable hypergraph families and their extremal structures.

Abstract

Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the diagonal tensor of degrees and adjacency tensor of $G$, respectively. In this paper, we establish a general theorem that extends the spectral Turán result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] to the setting of signless Laplacian spectral Turán problems. We prove that for any family $\mathcal{F}$ of $r$-uniform hypergraphs that is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and whose extremal constructions satisfy certain natural assumptions, the signless Laplacian spectral Turán problem can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely characterize the signless Laplacian spectral extremal hypergraph for the Fano plane.

Figures (1)

• Figure 1: Fano plane

Theorems & Definitions (29)

• Theorem 1.1: K.C.Chang.etc:Perron-Frobenius Theorem
• Theorem 1.2: Qi2013
• Definition 1.3: cooper2012spectra
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1: simonovits1968method
• Theorem 3.2: furedi2005triplekeevash2005turan
• Theorem 3.3: furedi2005triplekeevash2005turan
• proof
• proof