Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

TL;DR

This paper characterizes graphs that achieve specific regularity bounds for binomial edge ideals, linking graph properties like longest induced paths, cliques, and vertex counts.

Contribution

It provides complete characterizations of graphs satisfying certain regularity equalities involving induced paths, maximal cliques, and clique numbers.

Findings

Graphs with regularity equal to sum of longest induced paths and number of maximal cliques are characterized.

Connected graphs satisfying regularity equals longest induced path and a formula involving vertex count and clique number are characterized.

The range of possible regularity values within specified bounds is investigated.

Abstract

In this paper, we characterize all graphs $G$ satisfying \[\operatorname{reg}(S/J_G)=\ell(G)=c(G)\] where $\ell(G)$ is the sum of the lengths of the longest induced paths in each connected component of $G$ and $c(G)$ is the number of the maximal cliques of $G$. We also characterize all connected graphs $G$ that satisfy \[\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-\omega(G)+1\] where $\omega(G)$ is the clique number of $G$. Moreover, we investigate the possible values of the regularity of $S/J_G$ within the intervals $[\ell(G), c(G)]$ and $[\ell(G), |V(G)|-\omega(G)+1]$.