  In this paper, we use analysis on graphs to study quantitative measures of
segregation. We focus on a classical statistic from the geography and urban
sociology literature known as Moran's I, which in our language is a score
associated to a real-valued function on a graph, computed with respect to a
spatial weight matrix such as the adjacency matrix associated to the geographic
units that tile a city. Our results characterizing the extremal behavior of I
illustrate the important role of the underlying graph structure, especially the
degree distribution, in interpreting the score. In addition to the standard
spatial weight matrices encoding unit adjacency, we consider the Laplacian L
and a doubly-stochastic approximation M. These alternatives allow us to connect
I to ideas from Fourier analysis and random walks. We offer illustrations of
our theoretical results with a mix of stylized synthetic examples and real
geographic/demographic data.
