Three conjectures in extremal spectral graph theory

TL;DR

This paper proves three longstanding conjectures in extremal spectral graph theory, identifying unique graphs that maximize spectral invariants within specific graph families, using eigenvector-based structural analysis.

Contribution

It confirms three conjectures about spectral radius maximizers in planar, outerplanar, and connected graphs, providing new proofs and structural insights.

Findings

The join of P2 and Pn-2 uniquely maximizes spectral radius among planar graphs.

The join of a vertex and Pn-1 uniquely maximizes spectral radius among outerplanar graphs.

A pineapple graph uniquely maximizes spectral radius minus average degree among connected graphs.

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.