Dipole graph

In graph theory, a dipole graph, dipole, bond graph, or linkage, is a multigraph consisting of two vertices connected with a number of parallel edges. A dipole graph containing $n$ edges is called the size-$n$ dipole graph, and is denoted by $n ≥ 1$. The size-$n$ dipole graph is dual to the cycle graph $n ≥ 1$.

The honeycomb as an abstract graph is the maximal abelian covering graph of the dipole graph $Dn$, while the diamond crystal as an abstract graph is the maximal abelian covering graph of $Cn$.

Similarly to the Platonic graphs, the dipole graphs form the skeletons of the hosohedra. Their duals, the cycle graphs, form the skeletons of the dihedra.