Three conjectures in extremal spectral graph theory

Abstract: 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.