4 problems

Fix $d\in\mathbb{N}$ and let $H$ be a graph of maximum degree $3$. Let $k\in\mathbb{N}$, and let $G$ be an $(=\hspace{-0.2em}k,d)$-vertex expander, meaning that every set…

Let $\mathcal{G}$ be an infinite, irreducible, bounded-degree graph, let $\nu_{L,o}$ denote the $L$-step heat-kernel measure rooted at $o$, and let $\partial A$ be the boundary of…

Let ${\cal G}$ be a subgraph-closed class of graphs. Suppose that ${\cal G}$ has strongly sublinear separators. Subexponential expansion conjecture. The expansion of ${\cal G}$ is…

Let $G$ be a connected vertex-transitive graph, let $A\subseteq V(G)$ be finite with $0<|A|\leq \tfrac12|V(G)|$, let $\partial A$ denote the vertex boundary of $A$, and let…