14 problems

Let $G=(V,E)$ be a graph with degree bound $d_*$, and let $B(x,r)$ denote the open ball of radius $r$ about $x$. Assume the degree and curvature hypotheses referred to as Assumptio…

Let $G=(V,E)$ be a $5$-regular Ricci-flat graph. A graph is of Cartesian product type if it is the Cartesian product of a Ricci-flat $3$-regular graph and a Ricci-flat $2$-regular…

Bipartiteness and maximum-degree conjecture. 1. $G$ is bipartite. 2. For every edge $uv$,

Let $G^{(1)}$ and $G^{(2)}$ be graphs. For vertices indexed by $i$ and $j$, let ${\boldsymbol p}_i^{(1)}$ and ${\boldsymbol p}_j^{(2)}$ denote their node resistance curvatures, and…

Let $D\in\mathbb{N}$ be fixed. An increasing $D$-regular expander graph family is a sequence of $D$-regular expander graphs $\{G_k\}_{k\in\mathbb{N}}$ whose sizes increase with $k$…

Let $T_p$ be the infinite $p$-regular tree, and let $T_p^q$ be its $q$-fold Cartesian product, which is $pq$-regular. For a graph $G$ and vertex $x$, write…

Let $G=(V,E)$ be a strongly regular graph. Its girth is the length of its shortest cycle; in particular, girth $3$ means that $G$ contains a triangle. Strongly regular girth-three…

Let $d\in{\mathbb N}$. Non-expansion conjecture. No infinite family of finite, increasing, $d$-regular graphs satisfying $CD(0,\infty)$ can be a family of expander graphs. This is…

Let $d_{\max}\in{\mathbb N}$. A graph $G=(V,E)$ has vertex degrees $d_x\leq d_{\max}$ for all $x\in V$ and satisfies $CD(0,\infty)$. Bishop comparison conjecture. There are constan…

Let $G$ be a large but finite graph with negative curvature. Suppose its symmetric group fixes a unique point $v_0$. Let the inertia and demand of a vertex be the quantities define…

Let $G$ be a large but finite graph with positive curvature. A graph is vertex-transitive when for every pair of vertices there is a graph isomorphism taking one to the other. Let…

Let $G$ be a large but finite graph with positive curvature. Let the demand and inertia of a vertex be the graph quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov…

Let $G$ be a large but finite graph with negative curvature. Let the demand and inertia of a vertex be the graph quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov…

Let $G$ be a large but finite graph with negative curvature. Let the demand of a vertex mean the quantity measuring how many shortest paths pass through it, as defined in the paper…