Signless Laplacian Spectral Radius and Link Homology of Simplicial Complexes

Abstract

In this paper, we study the signless Laplacian spectral radius of pure simplicial complexes under local homological restrictions on links. Let $K$ be a pure $r$-dimensional complex on $n$ vertices, ${\mathfrak q}_{r-1}(K)$ be the spectral radius of the $(r-1)$-up signless Laplacian of $K$, and ${\operatorname{lk}}_K(σ)$ be the link of a face $σ$ in $K$. We prove that if the homology $\widetilde H_t({\operatorname{lk}}_K(σ), {\mathbb R})=0$ for every face $σ\in K$ with $|σ|=r-t$, then \[ {\mathfrak q}_{r-1}(K)\le tn-(t-1)(r+1).\] Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\binom{r+1}{t}\binom{r}{t}$, equality holds if and only if $K \cong Δ_{r+1-t} \star Δ_{n-r-1+t}^{t}$, where $Δ_n$ denotes a simplex on $n$ vertices, $Δ_n^{p}$ denotes the $(p-1)$-skeleton of $Δ_n$, and $\star$ denotes the join of two complexes.