8 problems

For integers $n$, $m$, and indices $i$, define … Thus $S_{\{i,j\}_n^m}$ is the multiset of non-negative integers no larger than $n$, with $m$ occurring twice, excluding $i$ and $j$…

Let $G$ be a graph on $n$ vertices, and let its Laplacian spectrum be the multiset of eigenvalues of its Laplacian matrix. $S_{n,n}$ conjecture. There exists no graph on $n$ vertic…

Selberg's conjecture. The smallest non-zero Laplacian eigenvalue is bounded below by

Laplacian limit-point conjecture. Every point in the interval

For integers $n$, $m$, and $i$ with $1\leqslant i<n$, let $S_{\{i,n\}_n^m}$ denote the multiset obtained from $\{0,1,\ldots,n\}$ by deleting $i$ and $n$, and repeating $m$ once, wi…

Let $S_{i,n}=\{0,1,2,\ldots,n\}\setminus\{i\}$, where the element $0$ is retained and the set has $n$ elements counting multiplicities. A set is Laplacian realizable if it is the L…

Let $\omega = \frac{1}{3}\left((19 + 3\sqrt{33})^{\frac{1}{3}} + (19 - 3\sqrt{33})^{\frac{1}{3}}+1 \right)$, and let $\rho_L(G)$ denote the Laplacian spectral radius of a graph $G$…

Generalized Brouwer conjecture. For every $t \in \mathbb{N}$,