26 problems

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

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 $G$ be a connected simple graph, and let $L(G)$ denote its Laplacian matrix. The critical group of $G$ is the torsion subgroup of the cokernel of a reduced Laplacian matrix of…

Let $m>9$ for the HKM difference sets and $m>7$ for the Lin difference sets, and let $\text{epsilon}(m)$ and $\text{delta}(m)$ be the quantities appearing in the formulas for the n…

Spin Smith normal form conjecture. The Smith normal form of $Z_{s,p}$ should be

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…

For each graph family in the paper, let the displayed symbols $M$, $A$, and their decorated variants denote the associated Kasteleyn-Percus or Kasteleyn matrices. Let round and $q$…

A Jacobi-Trudi matrix is a determinant matrix expressing a symmetric-function or enumeration formula in terms of complete symmetric functions; a $q$-specialization means that its e…

The paper considers Kasteleyn and Kasteleyn-Percus matrices associated with the listed planar graphs and their symmetry and $q$-weighted variants. Here, a matrix has a round Smith…

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…

Frost–Storey conjecture. The matrix $A$ is equivalent to its Smith normal form if and only if the reduced minors of every order generate the unit ideal in the polynomial ring.

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…

Alfaro–Zhou conjecture. Trees are determined by the spectra of $D^{\deg}_+$, $D^{\deg}$, and $A^{\operatorname{trs}}$, and are also determined by the Smith normal forms of…

Abiad–Alfaro conjecture. The SNF of $D^L$ and the SNF of $D^Q$ each determine every tree; equivalently, no two non-isomorphic trees have the same SNF for either matrix.

A tree is a connected acyclic graph, and the Smith normal form (SNF) of an integer matrix is its diagonal form under multiplication on the left and right by unimodular integer matr…

Let $\mathcal{A}$ be an Adinkra with Laplacian $L(\mathcal{A})$, $N$ colors, and $v$ vertices. Fix a color and form $\hat{L}(\mathcal{A})$ over $\mathbb{Z}[x]$ by replacing the dia…

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…

SNF determination conjecture for trees. Almost all trees are determined by the SNF of their distance Laplacian, and, analogously, by the SNF of their distance signless Laplacian.

Let $f$ be a bent function on $\mathbb F_2^n$. Write its Smith normal form as … Here $d_i$ are the elementary divisors and $m_i$ their multiplicities. Symmetry conjecture. The elem…

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…

Let $A$ be the adjacency matrix of the $n$-cube, and let an elementary divisor mean a diagonal invariant factor in the Smith normal form of $A$. For an integer $i\geq 0$, consider…