Periodic networks of fixed degree minimizing length

We study networks in ${\mathbb{R}}^n$ which are periodic under a lattice of rank~$n$ and have vertices of prescribed degree $d\ge 3$. We minimize the length of the quotient networks, subject to the constraint that the fundamental domain has $n$-dimensional volume~$1$. For $n=3$ and degree $3\le d\le 6$ we determine the minimizing networks with the least number of vertices in the quotient, while for $d\ge 7$ we state a length estimate. For general $n$, we determine the unique minimizers with $d=n+1$ and $d=2n$.