  We consider pairwise interaction energies and we investigate their minimizers
among lattices with prescribed minimal vectors (length and coordination
number), i.e. the one corresponding to the crystal's bonds. In particular, we
show the universal minimality -- i.e. the optimality for all completely
monotone interaction potentials -- of strongly eutactic lattices among these
structures. This gives new optimality results for the square, triangular,
simple cubic (SC), face-centred-cubic (FCC) and body-centred-cubic (BCC)
lattices in dimensions 2 and 3 when points are interacting through completely
monotone potentials. We also show the universal maximality of the triangular
and FCC lattices among all lattices with prescribed bonds. Furthermore, we
apply our results to Lennard-Jones type potentials, showing the minimality of
any universal minimizer (resp. maximizer) for small (resp. large) bond lengths,
where the ranges of optimality are easily computable. Finally, a numerical
investigation is presented where a phase transition of type
"square-rhombic-triangular" (resp. "SC-rhombic-BCC-rhombic-FCC") in dimension
$d=2$ (resp. $d=3$) among lattices with more than 4 (resp. 6) bonds is
observed.
