1) If n and k are odd positive integers with k<=n-1, then there are no graphs G such that G is k-regular with order n.
2) If n is even, k is a positive integer such that k<=n-1, then there are k-regular graphs with order n.
1) By contradiction. If G is a k-regular graph with order n, then the sum of ...
Statements involving k-regular graphs are proven.