How to computationally describe the fascinating diversity of indigenous languages of the Americas? This work provides a large-scale and quantitative approach to these languages through the proposal of a novel measure of linguistic complexity based on co-occurrence graphs. Linguistic complexity is measured using the set of eigenvalues obtained from the Laplacian matrix of graphs. The results suggest first that our graph-based definition of linguistic complexity is positively correlated with previous approaches. Second, we were able to describe some structural differences between indigenous languages. We argue that our approach might suggest another application of graph-based techniques to the study of language.