What is the minimum degree of a four-regular graph of diameter 2.
From trial and error I have concluded that the minimum degree is six. Explanation: If I start from one node then there must be 4 edges off this and each one of these edges must have a node attached to it, this gives us 5 nodes. Since the graph has diameter 2 I have to add a sith node to achieve this. Connecting these nodes with various edges will result in them all having degree four.
How would one rigorously justify this though?