14 problems

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…

For a connected undirected graph $G$, let $T(G;x,y)$ denote its Tutte polynomial and let $\EuScript{K}(G)$ denote its critical group. Non-determination conjecture. There exist conn…

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 $c\log(n)/n<p(n)<1-o(\log(n)/n)$. Square-free…

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

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 $a_1=1$ and $a_n=a_{n-1}^2+a_{n-1}$ for $n\geq 2$. For an arithmetical structure on the star graph $S_n$ or the complete graph $K_n$, its critical group is a finite abelian gro…

Let $a_1=1$ and $a_n=a_{n-1}^2+a_{n-1}$ for $n\geq 2$. For an arithmetical structure on the star graph $S_n$ or the complete graph $K_n$, let $\mathcal{K}$ denote its critical grou…

Let $B$ be an $m\times n$ matrix with integer entries. Write $D_k(B)$ for its $k$th determinantal divisor and $D_k^*(B)$ for the corresponding complementary determinantal divisor.…

Let $G$ be an arithmetical structure with associated quantities $\boldsymbol{d},\boldsymbol{r}$ and let $G'$ be obtained by the generalized star-clique operation, with associated q…

Let the $N$-cube be the graph whose vertices are $\mathbb{F}_2^N$, with edges joining vertices that differ in one coordinate, and let its critical group be the torsion part of the…

Clancy–Leake–Payne conjecture. As $n$ tends to infinity, the probability that $K(G(n,q))_{(p)}$ with its monodromy pairing is isomorphic to $(\Gamma,\langle\cdot,\cdot\rangle)$ is

Clancy–Leake–Payne conjecture. As $n$ tends to infinity,

Let $A$ be an abelian group of order $r$, let $\mathfrak{S}$ denote the tower of symmetric groups, and consider the differential tower of groups $A\wr\mathfrak{S}$. For $k\leq n$,…