43 problems

Grone–Merris conjecture. The eigenvalue sequence of $\Delta_0$ is majorized by the conjugate degree sequence:

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…

Let $G$ be a graph with Laplacian eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$, and let $d_k'$ be the number of vertices of $G$ having degree at least $k$. The sequence…

Zhou's distance Laplacian conjecture. For every integer $r$,

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…

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

Let $\mathbf{L}^{\rm spat}$ and $\mathbf{L}^{\rm temp}$ denote the spatial and temporal components of the inflated dynamic Laplacian, let $a$ be the coupling parameter, and let…

Let $T_1$, $T_2$, and $T_3$ be the three Penrose tilings considered in the paper, and let $\operatorname{dim}_{\mathrm{B}}$ and $\operatorname{cap}$ denote box-counting dimension a…

Let $T_1$, $T_2$, and $T_3$ be the three Penrose tilings considered in the paper, and let $\operatorname{Sp}(\Delta_{T_i})$ denote the spectrum of the graph Laplacian of $T_i$. Pen…

Let $\Omega$ be the continuum domain, let $\rho$ be the sampling density, and let $\Gamma$ be a set of continuum labels with label function $g$. Define $\overline g$ as the centere…

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 $G$ be the straight linear $3$-tree with $n$ vertices, and let $H$ be the straight linear $3$-tree with $n+1$ vertices. Write $r_G(1,n)$ and $r_H(1,n+1)$ for the total resistan…

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 $\bm L_{P}$ be the unweighted Laplacian matrix of the Petersen graph, which is regular with degree sequence $\bm d=\operatorname{diag}(\bm L_{P})=(3,3,\ldots,3)$. Let $\bm L^*$…

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…

Caputo–Knippel's no-solution conjecture. Under these assumptions, there are no solutions to this eigenvalue problem.

Caputo–Knippel's soft-node attachment conjecture. If $bbda-bbda'=p$ is an integer, then, although connecting $p$ soft nodes of $G$ to $G'$ is a possibility, there are no other poss…

Caputo–Knippel's converse conjecture. For the converse problem, $G$ and $G”$ should be connected by two of the simple transformations described above.

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