  How to obtain a graph from data samples is an important problem in graph
signal processing. One way to formulate this graph learning problem is based on
Gaussian maximum likelihood estimation, possibly under particular topology
constraints. To solve this problem, we typically require iterative convex
optimization solvers. In this paper, we show that when the target graph
topology does not contain any cycle, then the solution has a closed form in
terms of the empirical covariance matrix. This enables us to efficiently
construct a tree graph from data, even if there is only a single data sample
available. We also provide an error bound of the objective function when we use
the same solution to approximate a cyclic graph. As an example, we consider an
image denoising problem, in which for each input image we construct a graph
based on the theoretical result. We then apply low-pass graph filters based on
this graph. Experimental results show that the weights given by the graph
learning solution lead to better denoising results than the bilateral weights
under some conditions.
