Critical ideals of trees

Given a graph $G=(V, E)$, its generalized Laplacian matrix is given by $$L(G,X_G)_{u,v}= \begin{cases} x_u&\text{if }u=v,\\ -m_{uv}&\text{if }u\neq v, \end{cases}$$ where $X_G=\{x_u\, | \, u\in V(G)\}$ is a set of indeterminates and $m_{uv}$ is the number of edges between $u$ and $v$. The $j$-critical ideal of $G$ is the determinantal ideal generated by the minors of size $j$ of $L(G, X)$. A $2$-matching of $G$ is a subset $\mathcal{M}$ of its edges such that every vertex of $G$ has at most two incident edges in $\mathcal{M}$. We give a combinatorial description of a set of generators of the $j$-critical ideal of a tree $T$ as a function of a set of special $2$-matchings, which we called minimal, of the graph $T^\ell$ obtained from $T$ by adding a loop at each of its vertices. Also, we prove that the algebraic co-rank of $T$ is equal to the $2$-matching number of $T$, the maximum number of edges of a $2$-matching of $T$. As a consequence, one can compute each invariant factor of the critical group of any graph $G$ such that $G\setminus v$ is a tree for some of its vertices $v$, as the greatest common divisor of the evaluation of some polynomials associated to the minimal $2$-matchings of $T^\ell$. For instance, in the regular case, we recover some of the results obtained by Levine and Toumpakari about the critical group of a wired regular tree. Additionally, we can prove that the path $P_n$ is the unique simple graph with $n$ vertices and $n-1$ trivial critical ideals. We conjecture that the set of generators that we give is a reduced Gr\"obner basis and we can prove this for the $|V(T)|-1$-critical ideal. Finally, we apply the result in order to calculate the critical ideals of trees with depth two and some arithmetical trees associated to the reduction of elliptic curves of Kodaira type $I_n^*$.