8 problems

Fix a simple graph $G$ with minimum degree $\delta(G)$. A faithful orthogonal representation assigns vectors to the vertices so that two vertices are represented by orthogonal vect…

Reality-and-inequalities conjecture. The set of spectra realizable by nonnegative matrices is solvable with the reality condition and $\lfloor (n+1)/2\rfloor$ unions of $n$ polynom…

Basic semialgebraicity conjecture. The set of spectra realizable by symmetric nonnegative matrices forms a basic semi-algebraic set.

Let $G$ and $H$ be graphs such that $q(G)=q(H)=2$, with $MB(G)=k_1$ and $MB(H)=k_2$, and suppose that the quantities denoted in the source by $\mrr+(G)$ and $\mrr+(H)$ satisfy…

Let $G$ and $H$ be two graphs with $q(G)=q(H)=2$ and $MB(G)=MB(H)$. Join multiplicity-bipartition conjecture. One should have $MB(G\vee H)=MB(G)$, and this should hold in a more ge…

Let $T=S^k_{m,n}$ be the tree formed by taking a path on $k$ vertices and adding $m$ leaves to one endpoint and $n$ leaves to the other, where $kin\{2,3,4\}$ and $m,n>1$. Write…

Let $T=S^k_{m,n}$ be a type-one tree formed from a path on $k$ vertices, with $k e 2,3,4$ not assumed here, and let $[?]$ denote the relevant parameters. The complement $[?]$ is ob…

Let $T$ be a tree with $k$ high degree vertices, whose degrees are $d_1,\ldots,d_k$, and let $\mathcal{L}(T)$ denote the set of unordered multiplicity lists of real symmetric matri…