Erdős–Gallai theorem (statement)

Theorem. A non-increasing sequence $d_1\ge d_2\ge\cdots\ge d_n$ of nonnegative integers is graphic iff $\sum d_i$ is even and, for every $k\in\{1,\ldots,n\}$,
$$\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^{n}\min(d_i,k).$$

Conclusion. The theorem gives an algorithmic test for graphical sequences.