讲座内容：

Let X be a pure K+1-dimensional simplicial complex with orientation k, and s1 the k-up Laplacian spectral radius. In this talk, we investigate the upper and lower bounds on s1. On the one hand, we give an upper bound on s1. Moreover, if X is k+1-path connected, then characterize equality case. This generalizes some bounds on graph Laplacian to higher dimensions. On the other hand, we give a new lower bound on s1. This improves the lower bound given by Duval and Reiner (2002). As a corollary, we also derive a lower bound for s1 in terms of the k+1-diameter d of X, which not only strengthens the previously known bound for graphs (i.e., the case k=0), but also generalizes it to higher dimensions.