38 problems

Let $T$ be a tree with diameter $d$, and let $m_T[0,1)$ denote the number of Laplacian eigenvalues of $T$ less than $1$. Almost-all-trees lower-bound conjecture. Almost all trees h…

Let $T$ be a tree of order $n$, let $d$ denote its diameter, and let $m_T[0,1)$ be the number of Laplacian eigenvalues of $T$ in the interval $[0,1)$. Define … where…

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 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 $T$ 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…

Pólya's conjecture. The eigenvalue counting function of the Dirichlet Laplacian on $\Omega$ can be estimated from above by the leading term of Weyl's law, while the eigenvalue coun…

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…

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

Let $G$ be a connected graph of order $n$, let $d$ be its diameter with $d\ge 2$, and let $m_GI$ denote the number of Laplacian eigenvalues of $G$ in an interval $I\subseteq[0,n]$.…

Let $T$ be a tree with $n\geq 3$ vertices. The Laplacian spectral ratio is … where $\mu_1(G)$ and $\mu_{n-1}(G)$ are the largest and second smallest Laplacian eigenvalues of a conn…

Let $(M^m,g)$ be a smooth complete simply-connected Riemannian manifold with sectional curvature at most $\kappa$, where $\kappa\leq 0$. For a bounded domain with smooth boundary i…

Abrishami's conjecture. $G$ and $H$ have at least

Let $G$ be a simple graph with minimum degree $\delta$. Let $\mu_2$ and $\mu_n$ denote the second-smallest and largest eigenvalues of the Laplacian matrix of $G$, respectively, and…

A totally positive algebraic integer is an algebraic integer all of whose algebraic conjugates are positive real numbers. Totally positive algebraic integer Laplacian-eigenvalue co…

Let $\mu$ be a Laplacian eigenvalue of some tree $T$, with $\mu\neq 1$. Let $\mathcal{LT}_{\mathrm{min},\mu}^{0}$ and $\mathcal{LT}_{\mathrm{min},\mu}^{1}$ denote the two classes o…

Let $G$ be a graph of order $n\geq 2$, and let $\mu_1(G)\geq\cdots\geq\mu_{n-1}(G)$ be its Laplacian eigenvalues. The Laplacian spread is $\mu_1(G)-\mu_{n-1}(G)$. Zhai–Shu–Hong con…

Let $M$ be a complete, contractible Riemannian manifold, and let a group $\Gamma$ act properly, effectively, and cocompactly on $M$ by isometries. A Dirichlet domain of $\Gamma$ is…

Equal-area subspectral rectangles conjecture. There exists $s>0$ such that, for every $0<\epsilon<s$, $R_\epsilon$ is Dirichlet-superspectral to $R$ and Neumann-subspectral to $R$.…

Let $M$ be a normal manifold, and let $\{\Gamma_i\}$ be a finite open cover of $M$ by normal manifolds. Give the interior boundary components Neumann conditions. The finite-cover s…

Let $\Omega\subset\mathbb{R}^n$ be compact with Lipschitz boundary, and let $w$ be the single-term Weyl function of $\Omega$. The Dirichlet Laplace operator is superspectral to $w$…

Let $G$ be a simple graph of order $n\ge 2$, with Laplacian eigenvalues … The Laplacian spread conjecture. The inequality … holds, or equivalently … Moreover, equality holds if and…

Let $k$ be a positive integer, and let the $k$-uniform loose path of length $3$ be the hypergraph considered in Theorem. The conjecture. Theorem is true for the $k$-uniform loose p…

Let $G$ be a finite simple graph with Laplacian eigenvalues $\mu_1\geq\cdots\geq\mu_n$ and average degree $\overline{d}(G)=2m/n$, where $m$ is the number of edges and $n$ the numbe…