The critical group of a line graph

TL;DR

This paper explores the structure of the critical group of a line graph, providing new bounds and relationships based on graph properties, with implications for understanding spanning forests and algebraic invariants.

Contribution

It introduces three structural results on the critical group of line graphs, linking graph cycles, prime factors, and regularity to the critical group's structure.

Findings

Number of cycles bounds the critical group's generators

Constraints on p-primary structure based on degree sums

Exact sequences relating critical groups of graphs and their line graphs

Abstract

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph. The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices. The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups. The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion.