Signless Laplacian Spectral Radius and Link Homology of Simplicial Complexes

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 ${\mathrm{lk}}_{}(\sigma )$ be the link of a face $\sigma$ in $K$. We prove that if the homology ${\tilde{H}}_t({\mathrm{lk}}_{}(\sigma ),\mathbb{R})=0$ for every face $\sigma \in K$ with $\mathrm{\mid }\sigma \mathrm{\mid }=r-t$, then $${\mathfrak{q}}_{r-1}(K)\le tn-(t-1)(r+1)\mathrm{.}$$ Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\left(\genfrac{}{}{0px}{}{r+1}{t}\right)\left(\genfrac{}{}{0px}{}{r}{t}\right)$, equality holds if and only if $K\cong {\mathrm{\Delta }}_{r+1-t}\star {\mathrm{\Delta }}_{n-r-1+t}^t$, where ${\mathrm{\Delta }}_n$ denotes a simplex on $n$ vertices, ${\mathrm{\Delta }}_n^p$ denotes the $(p-1)$-skeleton of ${\mathrm{\Delta }}_n$, and $\star$ denotes the join of two complexes.