Largest-order critical group conjecture for arithmetical structures on star and complete graphs

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 group and let $|\mathcal{K}|$ denote its order. The construction considered has critical-group order $3a_{n-2}^2$. Largest-order critical group conjecture. For $n\geq 6$, the largest order of a critical group of an arithmetical structure on $S_n$ or on $K_n$ is

$$3a_{n-2}^2.$$

The preceding theorem gives the upper bound $|\mathcal{K}|<\frac{n!}{2}a_{n-2}^2$, while the displayed arithmetical structure realizes order $3a_{n-2}^2$. The conjecture asserts that this construction is optimal for all $n\geq 6$.