Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes

Let $K$ be a finite pure ridge-connected $d$-dimensional simplicial complex with at least two facets, and let $m_K(λ)$ denote the multiplicity of a positive eigenvalue $λ$ of its upper Laplacian $L_{d-1}^{\mathrm{up}}(K)$. We obtain upper bounds for $m_K(λ)$ in terms of the basic combinatorial param