Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2501.07494 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
Sommario:
- Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $λ_1 \ge λ_2 \ge \dots \ge λ_n$. The problem of bounding $λ_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov, who demonstrated strong upper and lower bounds for arbitrary $k$. Nikiforov also claimed a strengthened upper bound for $k \ge 3$, namely that $\frac{λ_k}{n} < \frac{1}{2\sqrt{k-1}} - \varepsilon_k$ for some positive $\varepsilon_k$, but omitted the proof due to its length. In this paper, we give a proof of this bound for $k = 3$. We achieve this by instead looking at $λ_{n-1} + λ_n$ and introducing a new graph operation which provides structure to minimising graphs, including $ω\le 3$ and $χ\le 4$. Then we reduce the hypothetical worst case to a graph that is $n/2$-regular and invariant under said operation. By considering a series of inequalities on the restricted eigenvector components, we prove that a sequence of graphs with $\frac{λ_{n-1} + λ_n}{n}$ converging to $-\frac{\sqrt{2}}{2}$ cannot exist.