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

TL;DR

This paper investigates the maximum signless Laplacian spectral radius of large $n$-vertex $r$-dimensional simplicial complexes without $r$-dimensional wheels, identifying extremal structures that achieve this maximum.

Contribution

It generalizes extremal spectral results from graphs to higher-dimensional simplicial complexes and provides a spectral analogue of a classical combinatorial theorem.

Findings

Identifies extremal complexes for large $n$ that maximize the spectral radius.

Generalizes known graph results to higher-dimensional complexes.

Provides a spectral analogue of a theorem by Sós, Erdős, and Brown.

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 S\'os, Erd\H{o}s and Brown on the maximum number of facets of simplicial complexes in the case $r=2$.