Coordination sequence

Summary

In crystallography and graph theory, the coordination sequence of a vertex $v$ is an integer sequence that counts how many vertices are at each possible distance from $v$. That is, it is a sequence

$$n_0, n_1, n_2,\dots$$

where each $n_i$ is the number of vertices that are $i$ steps away from $v$. If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of $v$. Coordination sequences can also be defined for sphere packings, by using either the contact graph of the spheres or the Delaunay triangulation of their centers, but these two choices may give rise to different sequences.