We discuss how to prove four conjectures in extremal graph theory where the graph invariant being maximized is a function of the eigenvalues or eigenvectors of the adjacency matrix of the graph. Our most difficult result proves a conjecture of Boots and Royle from 1991: the planar graph of maximum spectral radius is the join of an edge and a path of length $n-2$. All of our proofs follow a similar template, and we will end the talk with several problems to which one might try to apply our method. This is joint work with Josh Tobin.