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.