Graph theory problem: Given the diameter of a graph, and the amount of vertices stemming from each vertex. Prove an upper bound on the total number of vertices.

More specifically, you are given a graph that is of diameter = 2, and is 5-regular (5 vertices stemming from each vertex). Show that there are fewer than 26 vertices.

I know how to show there is an upper bound of 30 vertices, using a breadth first search tree (starting from initial node, adding 5 nodes, adding 5 nodes on to each of those nodes, to get diameter 2 and number of vertices =30). But I can't think of a way to show fewer than 26. Any sort of help would be greatly appreciated.