Date of Original Version
The Erdős–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.
Barrus, Michael D. The principal Erdos-Gallai differences of a degree sequence. Discrete Mathematics, vol. 345 (April 2022), no. 4, Article 112755. https://doi.org/10.1016/j.disc.2021.112755