5 problems

Complete-graph extremal conjecture. The number of arithmetical structures on $G$ is at most the number of arithmetical structures on the complete graph $K_n$:

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 $G$ 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…

Smooth-structure growth conjecture. The number of smooth arithmetical structures on $\mathcal{P}_{m,n}$ grows at the same rate as