14 problems

Let $G$ be a $d$-regular graph, and write $\mathrm{CD}(0,\infty)$ and $\mathrm{CD}(0,d)$ for the corresponding Bakry–Émery curvature-dimension conditions. Blachar–Pajot–Salez conje…

Regular-graph self-improvement conjecture. Any $d$-regular graph satisfying $\mathrm{CD}(0,\infty)$ satisfies $\mathrm{CD}(0,d)$.

Non-negative curvature implies polynomial growth. If $G$ has bounded degrees and satisfies $\mathrm{CD}(0,\infty)$, then $G$ has polynomial growth. More precisely,

Let $\kappa(\gamma)$ denote the limiting curvature constant associated with the fractional Laplacian parameter $\gamma$, as defined in the surrounding results. Positivity conjectur…

Let $G$ be a finite connected graph with minimum degree $\delta$ and non-negative Bakry–Émery curvature. A graph is $l$-edge-connected if deleting any set of fewer than $l$ edges l…

Let $G$ be a triangle-free graph satisfying $CD(0,\infty)$ with both the unweighted normalized and non-normalized Laplacians. An induced $k$-cycle conjecture. $G$ contains no induc…

Curvature flow convergence conjecture. The curvature flow converges for any initial condition $(G,P_0)$, that is, $(G,P(t))$ has a well-defined limit

Bakry–Émery curvature Bonnet–Myers conjecture. One has

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 $\Gamma$ have finite presentation … and let … be obtained by adding one defining relation. Assume that no generator in $S$ becomes redundant in $\Gamma'$: for distinct…

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 $G=(V,E)$ be a graph and let $x\text{ be a vertex}$. Let $G”'=(V”',E”')$ be obtained from $G$ by either deleting a leaf in $S_2(x)$ and its incident edge, or deleting…

Let $(N^n,g)$ be a complete Riemannian manifold with nonempty boundary, equipped with a smooth function $f$. Suppose that $N$ has nonnegative $m$-Bakry–Émery Ricci curvature and th…