Critical groups of arithmetical structures under a generalized star-clique operation

Abstract

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.

Figures (3)

• Figure 1: On the left is a graph $G$ with vertices labeled by $(d_i,r_i)$, the entries of an arithmetical structure $(\mathbf{d},\mathbf{r})$. Performing the operation at $v_7$ produces the graph $G'$ on the right with vertices labeled by entries of the arithmetical structure $(\mathbf{d}',\mathbf{r}')$.
• Figure 2: An arithmetical structure on a non-simple graph labeled by the entries of $(\mathbf{d},\mathbf{r})$ and the result of performing the operation at the vertex $v_4$.
• Figure 3: Another arithmetical structure on the same graph as in Figure \ref{['fig:example2a']} and with the same value of $d_4$ and the result of performing the operation at the vertex $v_4$.

Theorems & Definitions (27)

• Example 1
• Example 2
• Proposition 3
• proof
• Remark
• Proposition 4
• proof
• Lemma 5
• proof
• Proposition 6