Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

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 $[\ell(G), |V(G)|-\omega(G)+1]$.