Spectral radius of $2$-dimensional simplicial complexes with given Betti number

TL;DR

This paper derives an asymptotic formula for the signless Laplacian spectral radius of 2-dimensional simplicial complexes with specified Betti numbers and characterizes the complexes that maximize this spectral radius for certain Betti number values.

Contribution

It provides a new asymptotic formula for the spectral radius and characterizes extremal complexes based on Betti numbers, advancing spectral topology understanding.

Findings

Asymptotic formula for spectral radius established

Characterization of complexes with maximum spectral radius for Betti numbers 1 and 2

Insights into the relationship between Betti numbers and spectral properties

Abstract

In this paper we establish an asymptotic formula for the signless Laplacian spectral radius of a $2$-dimensional simplicial complex with given $2$-th Betti number. Furthermore, we characterize the $2$-dimensional simplicial complex that achieves the maximum signless Laplacian spectral radius among all-dimensional simplicial complex with the $2$-th Betti number equal to $1$ or $2$.