Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, chordal and very well-covered graphs

Rydberg blockade with multivalent atoms: effect of Rydberg series perturbation on van der Waals interactions

We investigate the effect of series perturbation on the second order dipole-dipole interactions between strontium atoms in $5sns({^1}S_0)$ and $5snp({^1}P_1)$ Rydberg states as a means of engineering long-range interactions between atoms in a way that gives an exceptional level of control over the strength and the sign of the interaction by changing $n$. We utilize experimentally available data to estimate the importance of perturber states at low $n$, and find that van der Waals interaction between two strontium atoms in the $5snp({^1}P_1)$ states shows strong peaks outside the usual hydrogenic $n^{11}$ scaling. We identify this to be the result of the perturbation of $5snd({^1}D_2)$ intermediate states by the $4d^2({^1}D_2)$ and $4dn's({^1}D_2)$ states in the $n<20$ range. This demonstrates that divalent atoms in general present a unique advantage for creating substantially stronger or weaker interaction strengths than those can be achieved using alkali metal atoms due to their highly perturbed spectra that can persist up to high-$n$