Graph limits and Hansen-Aouchiche conjecture
Organizer
Speaker
Lele Liu
Time
Wednesday, September 11, 2024 5:30 PM - 6:30 PM
Venue
Online
Abstract
Let $\lambda(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\overline{G}$ be the complement of $G$. The Hansen--Aouchiche conjecture states that the graph on $n$ vertices maximizing $\lambda(G) + \lambda(\overline{G})$ is the join of a clique and an independent set, with $\lfloor n/3\rfloor$ and $\lceil 2n/3\rceil$ (also $\lceil n/3\rceil$ and $\lfloor 2n/3\rfloor$ if $n \equiv 2 \pmod{3}$) vertices, respectively. We resolve this conjecture for sufficiently large $n$ using the theory of graph limits. In this talk, we will show how to use graph limits method to confirm this conjecture.