5 problems

The conjecture. The graph $\Gamma$ is unstable if and only if either there exists a nonzero $h\in 2\mathbb{Z}_{2m}$ such that

Structural conjecture. There exists $\alpha_0>0$ such that, for every $0\leq\alpha\leq\alpha_0$ and every $K_{r+1}$-free graph $G$ on $n$ vertices, there is an unbalanced blow-up…

Let $\Gamma$ be a graph. Its canonical double cover has adjacency matrix $\begin{pmatrix}0&A\\A&0\end{pmatrix}$ when $A$ is the adjacency matrix of $\Gamma$. The matrix $A$ is comp…

A circulant is a Cayley graph of a cyclic group; write an order-$n$ circulant as $\operatorname{Cay}(\mathbb{Z}_n,S)$. A circulant is nontrivially unstable when it is unstable but…

Let $G$ be a group and let $S$ be a nonempty inverse-closed subset of $G\setminus\{1\}$, where $1$ is the identity. The Cayley graph $\operatorname{Cay}(G,S)$ has vertex set $G$, w…