3 problems

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, let $\sigma(\mathcal{Q}(n))$ denote its spectrum, and let $\mathbb{Z}$ denote the integers. Integer-spectrum conjecture. For $n\geq4$…

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, and let $\gamma(\mathcal{Q}(n))$ denote its domination number. Cockayne's conjecture. For every $n \in \mathbb{N}$, … This asks wheth…

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, and let $\gamma(\mathcal{Q}(n))$ denote its domination number. For $k \in \mathbb{N} \cup \{0\}$, consider the order $n=4k+1$. Weakle…