The first one:
In this exercise I am asked to compute the number of 4-regular graphs of order 7.
I had an idea but I don't really know if it is the correct way of proceeding:
In a graph of order 7, the maximum degree is 6. Then, to find all 4-regular graphs, we can find all 2-regular graphs and take complementaries. However -I don't know if this is correct- in a 2-regular graph is a cycle, and all cycles of order 7 are isomorphic and thereby, there is only 1 4-regular graph of order 7.
The second one:
In this one, I am quite lost. I would like some ideas on how to start the proof.
Let G be a self-complementary graph whose order is congruent with 1 modulo 4, i.e. n = 4k + 1. I am asked to prove that there is an odd number of vertices of degree (n-1)/2, and of course, at least 1.
Thanks a lot! Any help is welcome :)