39 problems

Let $G=(V,E)$ be a graph, let $w\colon E\to\mathbb{Z}_+$ be an integer-valued edge-weight function, and let $\nu_w(G)$ and $\tau_w(G)$ denote, respectively, the maximum total weigh…

Let $U$ be a finite set, and let $\mu$ be a probability measure on the edges of the complete graph on $U$, so that $\mu(uv)\ge 0$ and … For $u\in U$, define the weighted degree by…

GHS23 divergence conjecture. If

Grigor'yan–Sun–Verbitsky conjecture. If

Let $K_n$ be the complete graph, let $e$ be an edge of $K_n$, and let $\boldsymbol{\dot{A}}(K_n)$ and $\boldsymbol{\dot{A}}(K_n-e)$ denote the degree-based weighted adjacency matri…

Consider the complete graph $K_n$ with vertices ordered $1,\ldots,n$, whose edges are ordered lexicographically and whose $i$-th edge has weight $r^{N-i}$, where $N=\binom{n}{2}$ a…

Judicious partition conjecture. Every weighted graph $G$ of order $n$ admits a $k$-partition $(X_1,X_2,\dots,X_k)$ satisfying both

Let $G=(V,E,w)$ be an edge-weighted complete graph on $n$ vertices, where $w:E\to[0,1]$. For $v\in V$, define its weighted degree by $d^w(v):=\sum_{uv\in E}w(uv)$ and let…

Let $G_\omega$ be a weighted tree. Proposition gives necessary conditions for $I(G_\omega)^n$ to be Cohen–Macaulay for all $n\geqslant 1$: $G$ has a perfect matching…

For $n\geq 2$, let $\mathcal{S}_n$ be the set of $n\times n$ real symmetric matrices with entries in $[0,1]$, and let $\lambda_{n-1}(M)$ denote the penultimate eigenvalue of $M$. W…

Let $\omega$ be sufficiently large and let $G=(V,E)$ be a graph with clique number $\omega$. Assign a vector $\vv_i\in\mathbb R^2$ to each vertex $i$, and define … for each pair of…

Higher-order Cheeger inequalities conjecture. There should hold related higher-order Cheeger inequalities for weighted graphs.

Minimum-degree conjecture. Every weighted undirected graph with minimum degree at least $k+1$ contains a cycle whose weight is divisible by $k$.

Fix integers $s$ and $t$ with $3\leq s\leq t-2$. A weighted graph admitting a $(b,t-1-b)$-partition has $b$ vertices and $t-1-b$ parts satisfying the defining partition conditions…

Weighted perfect-matching conjecture. There exists a perfect matching $M$ such that

Gutin–Yeo conjecture. Every weighted triangle-free subcubic graph has a cut of weight at least

Terai's conjecture. The ideal $I(G_{\mathbf w})$ is sequentially Cohen–Macaulay for every weight function $\mathbf w$. This conjecture extends questions about Cohen–Macaulayness an…

Weighted rainbow clique conjecture. If $n\ge r$, $G_0$ contains no $K_r$ with weight sequence bound $(1,2,\ldots,\binom{r}{2})$, then: (i) if $k=k_1$,…

Let $G$ be a graph, let $\phi$ be an edge-weighting, and let $D_{\phi G}$ denote the weighted distance matrix of $G$. For a positive integer $n$, let $K_{2,3,\ldots,3}$ denote the…

Let $G$ be a graph, let $m\geq 1$ and $k\geq 2$, and let $w$ be an $m$-weighting of $G$. A $k$-clique independent set is a set $I$ of vertices such that every $k$-clique of $G$ con…

Faster-than-polynomial decay conjecture. If the weights decay faster than polynomially, then the associated non-local derivatives should converge to the corresponding differential…

Complete-graph curvature flow conjecture. If $P_0$ is a non-degenerate Markovian weighting scheme without laziness on the complete graph $K_n$ with $n \ge 2$, then the curvature fl…

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

Let $(V,\b5)$ be a weighted graph, let $o\in V$, and let $\sigma>1$. A nonnegative solution $u$ of the elliptic inequality … is a function on $V$ satisfying this inequality, where…

SE$(s)$ extremal conjecture. The quantities $x$ and $y$ satisfy