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

TL;DR

This paper investigates how a generalized star-clique operation affects the critical groups of arithmetical structures on graphs, providing bounds and exact results for the transformed critical groups.

Contribution

It introduces bounds and exact formulas for the critical groups after applying a generalized star-clique operation on arithmetical structures.

Findings

Bounds the order and invariant factors of the new critical group.

Determines the critical group exactly for simple graphs.

Analyzes the transformation of critical groups under the 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.