Module hamming 

Hamming graph constructor (ALGO-CN-008).

Counterpart of igraph_hamming() in references/igraph/src/constructors/regular.c:1070-1153.

The d-dimensional Hamming graph H(n, q) over an alphabet of size q has q^n vertices indexed by all length-n strings in {0, 1, …, q-1}^n. Two vertices are connected iff their strings differ in exactly one position.

The string (x_0, x_1, …, x_{n-1}) maps to vertex id Σ_i x_i · q^i (little-endian base-q representation), matching the upstream convention.

Edge count: q^n · (q − 1) · n / 2.

Special cases:

• n = 0 → singleton (one vertex), even when q = 0.
• n > 0 and q = 0 → null graph (zero vertices).
• q = 1 → singleton (the only string is the all-zero one, no edges).
• q = 2 → identical to the n-dimensional hypercube Q_n (see crate::hypercube).

Algorithm: enumerate every vertex v ∈ [0, q^n); for each digit position i ∈ [0, n) extract dig = (v / q^i) mod q, then for every higher digit value dig + j with j ∈ [1, q − dig) emit the canonical edge (v, v + j · q^i). Because we only walk upward in each digit, every edge is produced exactly once (lower endpoint first). Mirrors the upstream C loop.

Time complexity: O(n · q^(n+1)) — for each of the q^n vertices we iterate n digit positions and emit up to q − 1 outgoing upward-pointing edges per digit.

Functions

hamming

Build the d-dimensional Hamming graph H(n, q).