  We prove three conjectures regarding the maximization of spectral invariants
over certain families of graphs. Our most difficult result is that the join of
$P_2$ and $P_{n-2}$ is the unique graph of maximum spectral radius over all
planar graphs. This was conjectured by Boots and Royle in 1991 and
independently by Cao and Vince in 1993. Similarly, we prove a conjecture of
Cvetkovi\'c and Rowlinson from 1990 stating that the unique outerplanar graph
of maximum spectral radius is the join of a vertex and $P_{n-1}$. Finally, we
prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph
is the unique connected graph maximizing the spectral radius minus the average
degree. To prove our theorems, we use the leading eigenvector of a purported
extremal graph to deduce structural properties about that graph. Using this
setup, we give short proofs of several old results: Mantel's Theorem, Stanley's
edge bound and extensions, the K\H{o}vari-S\'os-Tur\'an Theorem applied to
$\mathrm{ex}\left(n, K_{2,t}\right)$, and a partial solution to an old problem
of Erd\H{o}s on making a triangle-free graph bipartite.
