Degrees in graphs III: Which degrees sequences are possible?
Paul Erdős and Tibor Gallai developed a beautiful criterion to decide precisely when a degree sequence is graphical.
Degrees in graphs II: Degree sequences
The degree sequence of a graph is the sequence of degrees of all its elements.
Degrees in graphs I: the Handshake Lemma
A graph consists of objects called vertices and connections between them called edges. For every vertex, we can count how many neighbors it has, which is called its degree.