This is a graph theory question.

Definitions:

• Graphical Sequence: A sequence of numbers which works as a degree sequence in a graph.
• Ex, the graphical sequence of a straight line path of 4 vertices in a straight line path is {1, 2, 2, 1}, or {2, 2, 1, 1}, etc.

Let d1 d2 dn be a list of non-negative integers such that d1 n1, d1 dn 1 and

i=1ndi is even.

Prove that (d1, d2, . . . , dn) is a graphical sequence.