7 problems

Sharpness conjecture. For $e \in \mathbb N$ with $e \geqslant 3$ and $q$ a prime power, there exists $N \in \mathbb N$ such that for all $n \geqslant N$ we have equality in the cor…

Ghorbani's conjecture. This fraction tends to $0$ as $n$ tends to infinity.

Haemers' conjecture. The complete graph minimizes Seidel energy up to switching equivalence; equivalently,

Let $G_n$ be a simple graph on $V_n=\{v_1,\dots,v_n\}$, and let $S(G_n)$ be its Seidel matrix. Let $\det(S(G_n))$ denote its determinant. For a uniformly chosen graph on $V_n$, con…

Haemers' conjecture. The Seidel energy satisfies

Asymptotic sharpness conjecture. For all integers $e\geqslant 3$, there exists $N\in\mathbb N$ such that

Let $Q$ be a complex Seidel matrix whose off-diagonal entries are $m$th roots of unity for some $m\geq 3$. Such matrices are considered up to switching equivalence, where switching…