18 problems

Atiyah–Sutcliffe Conjecture 2. One has

Bourgain–Gamburd–Sarnak connectivity conjecture. The graph ${\mathcal{G}(p)}$ is connected for every prime $p>3$. This is the strong-approximation conjecture for the Markoff equati…

Let $G$ be a finite graph and define … Here $\partial S$ is the vertex boundary used in the paper, and $p^G$ is the percolation threshold. Isoperimetric-threshold conjecture. If…

Let $G$ be a finite transitive graph, and let the effective electric resistance between two vertices be computed in the graph's electrical network. Bounded-resistance conjecture. T…

Let $G$ be a finite transitive graph. Say that $G$ has the $EIT(c)$ property, for $0<c<1$, if for every pair of vertices $u,v\in G$ there is a set of paths from $v$ to $u$ equipped…

Let $G$ be a finite transitive graph, let $|G|$ denote its number of vertices, and let $p^G$ be its percolation threshold. Diameter criterion conjecture. There is a constant $C<1$…

Let a graph be better-quasi-ordered (BQO) under the minor relation when it satisfies the better-quasi-order property for that relation. Folklore conjecture. The finite graphs are b…

Let $T$ be a tree, meaning a connected circuit-free non-oriented finite simple graph, and let $C_T(\mathbb{R}^3)$ be its configuration space. Let $\mathcal{A}_T$ be the associated…

Let $G$ be a non-oriented finite simple graph, let $C_G(\mathbb{R}^3)$ be its configuration space, and let $\mathcal{A}_G$ denote the associated $G$-amplitude function. Graph-ampli…

Strengthened fixed-edge bound conjecture. For valency $4$, the bound $1/3$ in the classification theorem should be strengthened to $1/4$, eventually including some more small excep…

Benjamini's conjecture. The giant cluster should always be unique in the supercritical regime, irrespective of the geometry of $G$.

Let $\Gamma$ be a finite vertex-transitive graph. A graph is a $k$-multicirculant if it admits a cyclic group of automorphisms whose orbits on ${\rm V}(\Gamma)$ all have equal size…

Eigenvalue limit conjecture. If $p\equiv 1\pmod 4$, then $\lambda_2(p)$ converges to a limit $L$ satisfying

Let $\Gamma_n$ be the graph considered in the paper, and let $\mathcal{S}_n^\star$ be the distinguished update system supported on $\Gamma_n$, with preferred state $\star$. A point…

Bipartite-complement conjecture. The complement of $G$ is a bipartite graph.

Let $d\in\mathbb{N}$, and let $(\Gamma,G)\in\mathcal{A}(d)$ be a finite vertex-transitive graph of valency $d$ with a vertex-transitive automorphism group $G$. The pair is locally…

Let $c>0$. Let ${\mathcal F}_c$ be a finite family of graphs, let $G$ be an almost simple group, and let $\Gamma$ be a connected tetravalent $G$-arc-transitive graph with vertex…

Let $G$ be a finite graph on the vertex set $[d]=\{1,\ldots,d\}$ with $d\geq 7$, and let $K[G]$ be its edge ring. Depth-seven conjecture. … The claim arises from computations showi…