72 problems

Brouwer's conjecture. For every $k\in\{1,2,\dots,n\}$,

Let $M_g$ be a closed orientable surface of genus $g$. Let $\mathrm{chr}(M_g)$ be the supremum of the natural numbers $n$ for which the complete graph on $n$ vertices can be embedd…

Let $M$ be an embedded minimal surface in the unit sphere $\mathbb{S}^n$, and let $\lambda_1(M)$ denote its first non-zero Laplacian eigenvalue. Yau's conjecture. One has … The con…

Laplacian spread conjecture. For any graph $G$ of order $n\geq 2$,

Buser–Schmutz conjecture. The eigenvalue with index $2g+n-2$ satisfies

Let $G$ be a finite simple graph with $n$ vertices and $m$ edges. Let $\mu_1(G)\geq\mu_2(G)\geq\cdots\geq\mu_n(G)=0$ be its Laplacian eigenvalues, and define … A threshold graph is…

Let $G$ be a finite simple graph with $n$ non-isolated vertices. Let $L(G)$ be its Laplacian matrix, let $\lambda_1(G)\ge\cdots\ge\lambda_n(G)$ be the Laplacian eigenvalues, and wr…

Let $\mathbb{S}^2$ be the two-dimensional sphere, and let $\Lambda_k(\mathbb{S}^2)$ denote the supremum of the normalized $k$-th non-zero Laplace eigenvalue over metrics on…

Let $(M,\omega)$ be a closed symplectic manifold, where $\omega$ is its symplectic form. A quasi-Kähler structure on $(M,\omega)$ is a compatible almost complex structure and Riema…

Let $K$ be an $r$-dimensional simplicial complex, let $f_r(K)$ denote the number of $r$-faces, and let $\lambda_{r-1,i}(K)$ be the eigenvalues of the up-Laplacian on $(r-1)$-faces,…

Let $G$ be a graph with average degree $d 1$, and let $0=2_1(G) 2_2(G)\cdots 2_n(G)$ be its Laplacian eigenvalues. The Laplacian eigenratio of $G$ is $2_2(G)/2_n(G)$. Spielman's co…

Let $T$ be a tree on $n$ vertices, and let its combinatorial Laplacian eigenvalues be $\lambda_1,\dots,\lambda_n$. Their average is $2-\frac{2}{n}$. Small-eigenvalue conjecture. At…

Let $\Gamma=(G,\sigma)$ be a connected signed graph of order $n$. Let the Laplacian eigenvalues of $\Gamma$ be ordered as … and let its vertex degrees be ordered as … Hou–Li–Pan co…

Let $X$ be a simplicial complex on vertex set $V$. For $1\le r\le \dim(X)$, let $L_{r-1}^{+}(X)$ be the $(r-1)$-dimensional upper Laplacian, let $f_r(X)=|X(r)|$, and let…

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain. Let $\lambda_k$ and $\mu_k$ denote, respectively, the Dirichlet and Neumann eigenvalues of the negative Laplacian on…

Let $D=(0,\pi)\times(0,\pi)$, and consider the Dirichlet Laplacian eigenvalue problem … Its eigenvalues are $\lambda_{nm}=n^2+m^2$ for $n,m\in\mathbb{N}$, with multiplicity determi…

Let $T$ be a bounded triangle in the Euclidean plane, and let the Dirichlet eigenvalues of its Laplacian, listed with multiplicity, be … An equilateral triangle is a triangle whose…

Let $X_g$ be a closed hyperbolic surface of genus $g\geq 2$, and let $\lambda_1(X_g)$ denote the first positive eigenvalue of its Laplacian. Buser observed that arithmetic construc…

Let $\mathscr{D}\subset\mathbb{R}^d$ be a bounded Euclidean domain with $d\geq 2$, and consider the Dirichlet or Neumann Laplacian on $\mathscr{D}$. If…

Let … , and let … denote the eigenvalues of the Dirichlet and Neumann Laplacians on … be the semiclassical constant depending only on the dimension. Pólya's conjecture. For every o…

Let $G=(V,E)$ be a graph, let $\tau(G)$ denote its covering number, and let $L(G)$ be the Laplacian matrix of $G$. For $1\leq i\leq |V|$, write $\lambda_i(L(G))$ for the $i$-th lar…

Let $G=(V,E)$ be a graph with $n$ non-isolated vertices, and let $\nu(G)$ denote the maximum size of a matching in $G$. Let $L(G)$ be the Laplacian matrix of $G$, with eigenvalues…

Let ${\mathbb{RP}}^n$ be real projective space with round metric $g$, and let $wg$ be a conformally equivalent metric whose volume is normalized by … Write $\lambda_2(w)$ for the s…

Laplacian eigenvalue distribution conjecture. If $c=0,\dots,d-2$ and

Let $\Sigma$ be a closed, connected, smooth manifold of dimension at least $1$. For a Riemannian metric $h$ on $\Sigma$, let $\lambda_1(\Sigma,h)$ be the smallest nonzero eigenvalu…