55 problems

Dirac-support conjecture. The measure $\mu$ is minimizing for the $k$-th optimal eigenvalue:

Let $G$ be a non-circle graph and let $\Sigma_G$ be its secular manifold. The singular locus of the nonreduced variety $\Sigma_G$ has codimension at least one in $\Sigma_G$. Colin…

Let a graph be close to a tree, and let the nodal count of its $n$-th eigenstate be the number of its nodal domains. Nodal count conjecture. The nodal count of the $n$-th eigenstat…

Colin de Verdière's conjecture. The secular manifold $\Sigma(\Gamma)$ is irreducible if and only if $(\Gamma,\boldsymbol{\ell})$ admits no $\boldsymbol{\ell}$-independent isometrie…

Let $G$ be a graph, and let $\Gamma$ be the associated $\mathbb Q$-graph. For an integer $r\geq 1$, let $D(G,r)$ and $D(\Gamma,r)$ be the minimal degrees of a $g^r_d$ on $G$ and…

Inradius upper-bound conjecture. There is no universal constant $C>0$ such that

Mean-distance upper-bound conjecture. There is no universal constant $C>0$ such that

Star-graph critical-speed conjecture. For every $k\geq 3$ and every choice of edge lengths $\bm{\ell}$,

Let $G=(V,E,l)$ be a connected metric graph endowed with an interval exchange transformation $T$ partitioning $G$ into $n$ intervals with lengths $\ell_1, \ell_2, \ldots, \ell_n$ l…

Let $\Gamma$ be a metric graph of genus $g$, and let $d,r$ be integers satisfying $0<2r\leq d<g$. The Brill–Noether rank $w_d^r(\Gamma)$ is the tropical Brill–Noether rank. Jensen–…

Adaptability conjecture. The results of this work can be adapted to more general classes of metric graphs, such as single-knot metric graphs.

For each $r\geq 3$, let $B_r$ denote the lower-bound constant for rank-$r$ roses, and let $B_3$ be the rank-three constant. Monotonicity conjecture for $B_r$. … The claim would mak…

Let $\mathcal{R}_r$ denote the rank-$r$ rose, let $\mathcal{M}^1(\mathcal{R}_3)$ be the relevant space of normalized length functions, and let $B_3$ be the rank-three lower-bound c…

Let $G$ be a rank-$3$ graph with two vertices and four edges, and let $\ell=(\log(3),\log(3),\log(3),\log(3))$. Extremal graph conjecture. The most extreme case occurs for this gra…

For each rank $r\geq 3$, let $C_r$ be the constant from the theorem asserting that every rank-$r$ metric graph contains a proper subgraph whose entropy is at least $C_r$. Conjectur…

Optimal density constant conjecture. The density constant $\frac{1}{12}e(G)$ is not optimal, and $\frac{1}{8}e(G)$ should suffice.

Let $\mathcal G$ be a noncompact metric graph satisfying the so-called assumption (H), and let $\mu$ be any prescribed mass. Consider normalized solutions of the nonlinear Schrödin…

Let $\bm{G}_{g,L}$ be the random trivalent unicellular metric map of genus $g$ with boundary length $L$, and let $\bm{T}$ be the random infinite trivalent metric tree with independ…

Let $H$ be the Schrödinger operator on a finite metric graph with infinite ends, and suppose that the lower-bound assumption does not hold. Slower-decay conjecture. The…

Let $H$ be the Schrödinger operator on a finite metric graph with infinite ends, and let the initial datum be zero on the finite heads. Consider the third component of the correspo…

Let $p\in(2,6)$ and let $\nu>0$ be a mass. An energy ground state is a positive solution of the NLS variational problem on the $\mathcal{T}$-metric graph that minimizes the energy…

Let $Gamma$ be a compact metric graph with edge set $\mathcal{E}$. A Gaussian Markov random field of order $p$ is assumed to satisfy some mild regularity conditions. On each edge…

The paper considers Gaussian random fields on compact metric graphs, with Markov order measured through locality of the associated precision operator and isotropy defined using met…

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a domain, possibly equal to $\mathbb{R}^d$, and let $V:\Omega \to \mathbb{R}$ be a sufficiently smooth potential bounded from belo…

Fix a positive integer $r$. For a metric graph $\Gamma=(G,l)$ with rational edge lengths, the $r^{th}$ Metric Divisorial Gonality problem asks whether…