9 problems

Let $n$ be a positive integer, let a standardized Laplacian matrix of order $n$ be a matrix of the type considered in the paper, and let $S$ be the polygon defined earlier as the r…

Dalfó et al.'s conjecture. For any graph $G$ on $n$ vertices and any $1\leq k\leq n-1$,

Let the Laplacian matrix of a circle be formed using a choice of unit dual quaternion $\hat{\mathbf{q}}$. Conjecture. The eigenvalues of the Laplacian matrix of a circle are all no…

Let $G$ be a graph with adjacency matrix $A=A(G)$, and let $A_1=A(L_1(G))$ be the adjacency matrix of its line graph. Write $c_{n-k}(G)$ for the corresponding Laplacian coefficient…

Let $G$ be a simple graph on the vertex set $V=\{0,1,\ldots,n\}$ with root $0$, let $R=\mathbb K[x_1,\ldots,x_n]$, and let $\mathcal{M}_G^{(1)}$ be the $1$-skeleton ideal of its…

Let $\mathcal{G}$ be a connected undirected weighted graph, and vary its non-null edge weights while preserving its graph topology. Let $v$ be a Fiedler vector, namely an eigenvect…

The spanning-tree graph minor product conjecture. The polynomial $\Phi_G$ is a product of symmetric minors of $Q$, namely polynomials of the form $\det(Q_W)$. Equivalently, the num…

Let $T$ be a tree with vertex set $V(T)=\{1,\ldots,n\}$, and let $d_{ij}$ denote the distance between vertices $i$ and $j$. For a symmetric, nonnegative matrix $A=(a_{ij})$, define…

The polygonal spectral enclosure conjecture. All eigenvalues of every standardized Laplacian matrix $\widetilde L\in\widetilde\Lambda_n$ belong to $S(n)$. The conjecture would iden…