The regularity of edge rings and matching numbers

TL;DR

This paper establishes bounds on the regularity of edge rings of graphs based on their matching numbers, differentiating between bipartite and non-bipartite cases under normality conditions.

Contribution

It provides new upper bounds for the regularity of edge rings in relation to the matching number, considering the graph's bipartiteness and normality.

Findings

For non-bipartite graphs with normal edge rings, regularity is at most the matching number.

For bipartite graphs, the regularity is at most the matching number minus one.

The results connect algebraic properties of edge rings with combinatorial graph invariants.

Abstract

Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq \mat(G) - 1$ if $G$ is bipartite.