This project presents an innovative way of solving the inexact graph matching problem of weighted graphs. The goal is to find the correspondence between the vertices of two similar graphs. This problem is solved through a multi-resolution approach in the spectral domain. Spectral graph matching provides a framework that allows to represent both graphs in a different dimensional space where the problem is more tractable. Furthermore, the multi-resolution approach improves the performance of the graph matching algorithm, since lower resolutions reveal new structural patterns. To obtain lower resolutions of both graphs, the proposed graph downsampling methods must make sure that the vertices selected for the lower resolutions should be the ones that are more likely to be correctly matched. If this property is not accomplished, the matching in this lower resolution will not provide useful information for the final matching. A novel solution to common challenges in graph matching, i.e, sign ambiguity of the eigenvectors, are provided in order to improve the correspondence between the vertices of both graphs. Different graph matching applications are presented to illustrate the effectiveness of the proposed technique. A comparison with other spectral graph matching algorithms demonstrates the benefits of the proposed approach.