We will present theory and applications of a class of random graphs that we call entropic graphs. Such graphs are entropic in the following sense. The length functional of an entropic graph spanning N realizations of a Lebesgue density converges a.s. in N to the Renyi alpha-entropy of the density. For example, any graph that is continuous and quasi-additive as defined in Yukich (1998) is entropic. The consistent entropy estimation property of entropic graphs motivates their use in applications such as clustering, alignment of multidimensional data sets, and pattern matching. We present an application to registration of a pair of image volumes in medical image databases.