Performing direction-of-arrival (DOA) estimation of a detected source target using data collected from an array of sensors is a long-studied problem in signal processing. The emerging framework of signal processing on graphs (SPG) offers the opportunity to look at this problem from a different point of view by adding the concept of a graph underlying the measured data as an extra layer of information that can be incorporated in to the data processing system. While a recent presentation [J. Acoust. Soc. Am. 143, 1852 (2018)] showed how to use the Graph Fourier Transform to estimate DOA on a uniform line array, the purpose of this talk is to show how we can use the smoothness of a graph signal across vertices for the more involved case of a non-uniform line array. The scope of the problem was narrowed to DOA estimation of a single, far-field, and narrowband source signal. An iterative approach was developed that demonstrates the utility of SPG, and the approach was verified and tested through simulation under different signal-to-noise ratios. We also compared the results to conventional delay-and-sum beamforming to gain insight on what differentiates SPG over existing methods.