Signless Laplacian spectral radius of simplicial complexes without holes

TL;DR

This paper investigates the maximum signless Laplacian spectral radius of simplicial complexes without holes, extending classical graph extremal results to higher dimensions and providing bounds related to Betti numbers.

Contribution

It determines the structure of complexes that maximize the spectral radius and establishes upper bounds based on topological invariants, extending Turán-type results to simplicial complexes.

Findings

Identified the extremal simplicial complex structure for maximum spectral radius.

Established upper bounds on spectral radius using Betti numbers.

Connected spectral radius bounds to face numbers and Turán numbers.

Abstract

We study a spectral analog of the Tur\'an problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Tur\'an numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes.