A lower bound for the second largest up-Laplacian eigenvalue of a simplicial complex

Let $K$ be a finite $k$-dimensional simplicial complex with at least two $k$-faces, and let $\Lup_{k-1}(K)=\partial_k\partial_k^{\T}$ be its $(k-1)$-dimensional up-Laplacian. Writing $d_2(K)$ for the second largest upper degree of a $(k-1)$-face, we prove the sharp bound $$\lambda_2\bigl(\Lup_{k-1}(K)\bigr)\ \ge\ d_2(K)+k-1 .$$ For $k=1$, this recovers the graph inequality $\lambda_2(L(G))\ge d_2(G)$ of Li and Pan. We also show that the full graph inequality of Brouwer and Haemers does not extend directly to simplicial complexes: the natural candidate $\lambda_m(\Lup_{k-1}(K))\ge d_m(K)-m+k+1$ already fails for $m=3$.