Let $G$ be a finite simple graph with $n$ vertices and $m$ edges, and let $d_1 \le d_2 \le \dots \le d_n$ be its vertex degrees listed in nondecreasing order. The annihilation number $a(G)$ is the largest integer $k$ with $0 \le k \le n$ such that

that is, the largest $k$ for which the $k$ smallest degrees sum to at most the number of edges. The value $k = 0$ always qualifies, so $a(G)$ is well defined.

The annihilation number was originally defined through a reduction process on the degree sequence; the definition above is an equivalent characterization and is the one in common use.