3 problems

Let $A(n,k)$ be the $(n,k)$-arrangement graph, with $k$ fixed and $n$ sufficiently large. Chen–Ghorbani–Wong conjecture. The only negative eigenvalue in the spectrum of $A(n,k)$ is…

Let $A(n,k)$ be the arrangement graph whose vertices are the $k$-permutations of $[n]$, with two vertices adjacent when they agree in exactly $k-1$ positions. Eventual negativity c…

Let $A(n,k)$ be the arrangement graph whose vertices are the $k$-permutations of $[n]$, with two vertices adjacent when they agree in exactly $k-1$ positions. Integrality conjectur…