12 problems

Let $\mathsf{T}$ be the tree in Fig., and let $T$ be any nontrivial tree with root $r$. Define $G_1=T(r) \odot \mathsf{T}(4)$ and $G_2=T(r) \odot \mathsf{T}(7)$. Smith normal form…

For the explicitly defined $q$-weighted Kasteleyn and Kasteleyn-Percus matrices $M(a,b,c;q)$, $M_\rho(a,a,a;q)$, $A_\tau(a,b,b;q)$, $\widetilde{A}_\tau(a,b,b;q)$, and…

Let $\EuScript{G}(n,p)$ be a random simple undirected graph with $n$ vertices and edge-probability $0\leq p\leq 1$. For $c>1$, assume $c\log(n)/n<p(n)<1-o(\log(n)/n)$. Bounded-expe…

Let $G$ be the topological ordering of a graph on $[n]$, and let $B$ be the matrix over ${\mathbb{Z}}/p{\mathbb{Z}}$ given by … Let $r_p$ be the rank of $B$ over…

Smith normal form conjecture. The Smith normal form of $W(\tilde{D}_n)$ is

Let $D_n$ ($n\ge 4$) be the Dynkin graph, let $W(D_n)$ be its walk matrix, and let $\hat{W}(D_n)$ be the $(n-1)\times(n-1)$ matrix obtained from $W(D_n)$ by removing the first row…

Let $L$ be the Laplacian of an Adinkra, and let $\hat{L}$ be its polynomial lift over $\mathbb{Z}[x]$. Write the Smith normal form of $L$ as a tuple of invariant factors, with mult…

Let an Adinkra be a signed graph with signed Laplacian $L$, and let its invariant factors mean the diagonal entries in the Smith normal form of $L$. A signature is the choice of si…

Let $P$ be a differential poset, let $p_n$ denote the size of its rank-$n$ set, and let $U_{n-1}D_n$ be the composite of the down operator $D_n$ from rank $n$ to rank $n-1$ with th…

Daniel–Jolliffe conjecture. For every $i$, the multiplicity of $2^i$ as a $2$-elementary divisor of $A$ equals the number of eigenvalues of $A$ with exact $2$-power divisor…

Miller–Reiner conjecture. For any differential poset $P$ and any $n\geq 0$, the matrix

The map $k\frac{\partial}{\partial p_k}p_k$ acts on homogeneous symmetric functions of degree $n$. With respect to an integral basis, let $M_k^{(n)}$ denote its matrix, and let…