A spectral Moore problem and the Yang-Koolen conjecture

Abstract

We study the maximum order $v(k,\theta )$ of a connected $k$-regular graph whose second largest eigenvalue is at most $\theta$. For every real $\theta \ge 1$, we prove that \[ T(θ) =\limsup_{k\to\infty}\bigl(v(k,θ)-2k\bigr) =2\left\lfloor1+\frac{θ^2}{4}\right\rfloor. \] In particular, $T(\theta )>2\theta$ for every real $\theta \ge 4$, disproving the Yang-Koolen conjecture. We also determine $v(k,\theta )$ for all sufficiently large $k$ on explicit intervals of $\theta$. We prove a jump in the second largest eigenvalue above $1$. For each fixed $1\le \theta <2$ and all sufficiently large $k$, every connected $k$-regular graph of order at least $2k+2$ whose second largest eigenvalue is at most $\theta$ is the crown graph. For $\theta =\sqrt{2}$, we prove that $v(k,\sqrt{2})=2k+2$ for every $k\ge 13$. Finally, we obtain two new upper bounds on $v(k,\theta )$.

arXiv comment14 pages