networkx.algorithms.graphical.is_valid_degree_sequence_erdos_gallai¶

is_valid_degree_sequence_erdos_gallai
(deg_sequence)[source]¶ Returns True if deg_sequence can be realized by a simple graph.
The validation is done using the ErdősGallai theorem [EG1960].
Parameters: deg_sequence (list) – A list of integers Returns: valid – True if deg_sequence is graphical and False if not. Return type: bool Notes
This implementation uses an equivalent form of the ErdősGallai criterion. Worstcase run time is \(O(n)\) where \(n\) is the length of the sequence.
Specifically, a sequence d is graphical if and only if the sum of the sequence is even and for all strong indices k in the sequence,
\[\sum_{i=1}^{k} d_i \leq k(k1) + \sum_{j=k+1}^{n} \min(d_i,k) = k(n1)  ( k \sum_{j=0}^{k1} n_j  \sum_{j=0}^{k1} j n_j )\]A strong index k is any index where d_k >= k and the value n_j is the number of occurrences of j in d. The maximal strong index is called the Durfee index.
This particular rearrangement comes from the proof of Theorem 3 in [2].
The ZZ condition says that for the sequence d if
\[d >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}\]then d is graphical. This was shown in Theorem 6 in [2].
References
[1] A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”, Discrete Mathematics, 265, pp. 417420 (2003). [2] (1, 2) I.E. Zverovich and V.E. Zverovich. “Contributions to the theory of graphic sequences”, Discrete Mathematics, 105, pp. 292303 (1992).