Abstract: Spectral clustering is a popular unsupervised learning technique for finding meaningful structure in large datasets. A weighted graph is constructed on the dataset, encoding the similarities between the data points. A graph Laplacian operator is then defined on this graph whose spectral geometric content reveals the number and shape of clusters in the data set. In this talk I will present some spectral analysis of graph Laplacians in the continuum limit where the number of vertices of the graph goes to infinity. In the first part I will discuss how the different normalizations of the graph Laplacian will affect the spectrum of the continuum operator and introduce a notion of a balanced normalization that has desirable qualities in large data settings. In the second part of the talk I will focus on a specific choice of the graph Laplacian and present some results on the consistency of spectral clustering by first studying the continuum limit operator and extending its properties to discrete approximations.