23 problems

Non-degenerate-subgraph conjecture. The subgraph $G_W$ coincides with the subgraph of $G$ induced by $W$, all transition rates from $W$ to $V\setminus W$ are zero, and $(G_W,P_W)$…

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…

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…

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…

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

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…

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

Weighted Laplacian Spread Conjecture. One has

Let $G$ be a weighted triangle-free graph, and let $T$ be a spanning tree of $G$. Spanning-tree bound conjecture. One should have … The conjecture would determine the optimal value…

Let $D$ be an arc-weighted digraph without loops or two-cycles. For each vertex $v$, let $\delta_v$ be the second-neighborhood weight minus the first-neighborhood weight, as define…

Let $D$ be a digraph without loops or two-cycles. Give each vertex a nonnegative real weight, and call a vertex weakly expanding when the weight of its second out-neighborhood is a…

For integers $r\geq 2$ and $t\in(0,1]$, let $\delta(r,t)$ denote the infimum of the minimum weighted-degree density guaranteeing a partition into $r$-cliques of total weight at lea…

Let $G_w$ be a graph equipped with the minimum-degree weighting scheme, let $n$ be its number of vertices, and let $d$ be the parameter used in the locally tree-like analysis. Let…

Let $T$ be a tree, let $E(T)$ denote its edge set, and let $wp(T)$ be its weighted pebbling number. For a weighted graph, write $w_T(k)$ for the minimum total edge weight required…