Graph diameter ($d(G)$)

The diameter of a connected finite graph is the maximum distance between two vertices.

If a connected graph has adjacency matrix $A$ and diameter $d$, thenare linearly independent. For the largest index $k$ with a nonzero coefficient, choose vertices at distance $k$; the corresponding entries of all lower powers vanish, while $(A^k{)}_{uv}>0$ by the walk count from powers of an adjacency matrix.