Self Complementary Graph Proof

[Trentonx]

Homework Statement

Suppose [itex]G \cong\bar{G}[/itex] and that [itex]n = |V(G)|=4k+1[/itex] for some [itex]k \ge 1[/itex]. Suppose that the degree sequence of G is [itex]d_{1}\ge d_{2} \ge ... \ge d_{n} [/itex]

Prove that [itex]d_{i}+d_{n-i+1}=n-1[/itex] for each [itex]i=1, 2,...,n[/itex]

Homework Equations

I don't think there are any.

The Attempt at a Solution

I put the question in terms of k [itex]d_{i}+d_{4k-i+2}=n-1[/itex]
and found the number of edges, since the graph has to have exactly half the maximum.
[itex]|E(G)| = \frac{n(n-1)}{4}=\frac{(4k+1)(4k)}{4}=4k^{2}+4 [/itex]
I'm not sure that help's, but I thought it might be useful. I need something about the degree sequence, and how it will look, but I'm not sure where to get that.

 

[Nino MMP]

I tried to work out the degree sequence and did a proof by contradiction, but that didn't get me anywhere. Any help is appreciated.