18 problems

H-free Brouwer conjecture. For every $H$-free graph $G=(V,E)$ with $|V|\ge k$,

Let $G=(V,E)$ be a graph and let $F_k(G)$ be its $k$-th token graph, whose vertices are the $k$-element subsets of $V$. For $1\le k\le |V|/2$, let $L(F_k(G))$ denote the Laplacian…

Let $G$ be a graph on $n$ vertices. Let $a(G)$ denote its algebraic connectivity, and let $\omega(G)$ denote the minimum entry of its matrix of relative forest accessibilities. Mer…

Vietoris–Rips ultrametric condition. The Vietoris–Rips clusters for some $\epsilon>0$ realise the sufficient ultrametric condition

Let $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$. Consider the rescaled averaging Laplacian and the rescaled graph Laplacian at random sample points, both de…

Let $\mathrm{SG}$ be the Sierpiński gasket, let $\mathcal{L}$ be its Laplacian, and let $\mathcal{L}_\epsilon(\omega)$ be the random graph averaging Laplacian defined from random s…

Stability conjecture. Eigenmaps of the random averaging Laplacians give numerically stable locally uniform approximations to the corresponding eigenmap of $\mathcal{L}_{(g)}$. This…

Let $\phi_k$ denote the $k^\text{th}$ eigenfunction of the Laplacian on $[-1,1]$, and let $\phi_{\epsilon,n,k}$ denote the $k^\text{th}$ eigenfunction of a graph Laplacian…

Let a family of graphs have Laplacian matrices, and apply the procedures LaplaceExpand and SystemReduce together with the mathematical theorems described in the source. The numerat…

Consider a family of matrices to which the Laplace expansion process described in the source is applied. Convergence conjecture. For matrix families satisfying specified criteria,…

Let $n$ be odd, and let $P_n$ be a path graph with Laplacian matrix $L_n$. For 1-port selection, perturb $L_n$ to … where $p^*=(n+1)/2$, and denote the resulting optimal perturbed…

Let $G$ be a graph with minimum degree … denote the parameter introduced above. Assume that $G$ is not the cycle graph. Non-cycle bound. … This conjecture refines the preceding gen…

Decomposability conjecture. The graph $G$ of a $D$-optimal design in the Bradley--Terry model is decomposable.

Let $G$ be an undirected, connected multigraph with genus $g\geq 1$ and $n\geq 2$ vertices. Let $\mathcal{N}_G$, ${\rm Crit}_{\triangle}(L_G)$ and…

Let $\mathsf{G}$ be a graph and let $\mathsf{G}_D$ be the graph obtained by disaggregating a vertex of $\mathsf{G}$, with weights on the internal edges of the disaggregation. Kuhle…

Let $G=(V,c,E)$ be the one-dimensional weighted graph considered above, with edge conductances $c_{n,n-1}=a_n$ and energy Hilbert space $\mathscr{H}_{E}$. Let $\Delta$ denote the c…

Near-symmetry conjecture. The generating function has this form, and if $(a_0,\ldots,a_j)$ is the coefficient list of $f(q)$, then appending $0$ to the end of this list and subtrac…

Cyclically distinct composition conjecture. The number of integer points $\lambda\in\mathcal{C}_n$ with $\lambda_0=m$ is equal to the number of compositions of $m$ into $n$ parts t…