Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

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 \[\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-ω(G)+1\] where $\ddot{I}‰(G)$ 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 $[\ell (G),|V(G)|-\ddot{I}‰(G)+1]$.

17 pages, 3 figures