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Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels

Huan-Zhi Zhang, Yi-Zheng Fan

Abstract

An $r$-dimensional wheel is defined as the join of an $(r-2)$-simplex and a cycle. In this paper, we study the maximum signless Laplacian spectral radius of $n$-vertex $r$-dimensional pure simplicial complexes that contain no $r$-dimensional wheels. For sufficiently large $n$, we determine the extremal complexes that attain this maximum. Our result generalizes the corresponding extremal results of signless Laplacian on graphs and provides a spectral anlogue of a theorem of Sós, Erdős and Brown on the maximum number of facets of simplicial complexes in the case $r=2$.

Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels

Abstract

An -dimensional wheel is defined as the join of an -simplex and a cycle. In this paper, we study the maximum signless Laplacian spectral radius of -vertex -dimensional pure simplicial complexes that contain no -dimensional wheels. For sufficiently large , we determine the extremal complexes that attain this maximum. Our result generalizes the corresponding extremal results of signless Laplacian on graphs and provides a spectral anlogue of a theorem of Sós, Erdős and Brown on the maximum number of facets of simplicial complexes in the case .

Paper Structure

This paper contains 10 sections, 8 theorems, 32 equations, 2 figures.

Table of Contents

  1. introduction
  2. Turán problems
  3. Spectral Turán problems
  4. Preliminaries
  5. Simplicial complexes and Laplace operators
  6. Signless Laplace operators
  7. Higher dimensional wheels
  8. Cocycle complexs
  9. 1-neighbor uniformity
  10. Proof of main result

Key Result

Theorem 1.1

Let $K$ be a pure $r$-dimensional complex on $n$ vertices without $r$-dimensional wheels. Then Furthermore, if $K$ is $r$-path connected, then, for sufficiently large $n$, the equality holds in upp-main if and only if $K \cong {\sf{B}}_n^{r}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 3.1: A triangulation of Möbius strip
  • Figure 3.2: An illustration of Case 2 in Proof of Lemma \ref{['r+3']}

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 7 more