Let K be a pure \((i+1)\) -dimensional simplicial complex with orientation \(\sigma \) , and \(s_1\) the i-up Laplacian spectral radius. In this paper, we investigate the upper and lower bounds on \(s_1\) . On the one hand, we show that \(\begin{aligned} s_1\le \max \left\{ d^{+}_{S_{i+1}(K)}(E)+m(E):E\in S_{i}(K)\right\} , \end{aligned}\) where \(d^{+}_{S_{i+1}(K)}(E)\) is the upper degree of E and \(\begin{aligned} m(E)=\frac{\sum _{E^{\prime }\in S_{i}(K), E^{\prime }\cup E\in S_{i+1}(K) }d^{+}_{S_{i+1}(K)}(E^{\prime }) }{d^{+}_{S_{i+1}(K)}(E)}. \end{aligned}\) Moreover, if K is \((i+1)\) -path connected, then equality holds if and only if \(d^{+}_{S_{i+1}(K)}(E)+m(E)\) is a constant for any \(E\in S_{i}(K)\) , and \(\left( B_{i-1}(K),\sigma \right) \) is balanced. This generalizes some bounds on graph Laplacian to higher dimensions. On the other hand, we show that \(\begin{aligned} s_1\ge i+2+ \max \left\{ \frac{\sum _{F\in S_{i+1}(K^\prime ) }d^{-}_{S_{i+1}(K^\prime )}(F)}{|S_{i+1}(K^\prime )|}\right\} , \end{aligned}\) where the maximum is taken over all subcomplexes \(K^{\prime }\) of K such that \(\left( B_{i}(K^\prime ),\sigma _{K^\prime }\right) \) is balanced. This improves the lower bound given by Duval and Reiner (2002). As a corollary, we also derive a lower bound for \(s_1\) in terms of the \((i+1)\) -diameter d of K, which not only strengthens the previously known bound for graphs (i.e., the case \(i = 0\) ), but also generalizes it to higher dimensions.
