Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

In this paper, we characterize all graphs $G$ satisfying $$\mathrm{reg}(S\mathrm{/}{J}_G)=\mathrm{\ell }(G)=c(G)$$ where $\mathrm{\ell }(G)$ 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 $$\mathrm{reg}(S\mathrm{/}{J}_G)=\mathrm{\ell }(G)=\mathrm{\mid }V(G)\mathrm{\mid }-\omega (G)+1$$ where $\omega (G)$ is the clique number of $G$. Moreover, we investigate the possible values of the regularity of $S\mathrm{/}{J}_G$ within the intervals $[\mathrm{\ell }(G),c(G)]$ and $[\mathrm{\ell }(G),\mathrm{\mid }V(G)\mathrm{\mid }-\omega (G)+1]$.