TITLE: How many nodes are there in a 5-regular planar graph with diameter 2?
QUESTION [5 upvotes]: Undergrad here; I honestly have no idea what to do. I can't even imagine what a 5-regular graph with diameter 2 would look like, let alone a planar one.

REPLY [9 votes]: No such simple graph can exist, the smallest 5-regular planar graph is the icosahedron and it has diameter 3 (the distance between the green and yellow vertex is 3). I proved that there are actually only two simple planar 5-regular graphs with diameter less than 4 in my paper in Ars Combinatoria (Volume CVI, July, 2012). See my webpage about extremal regular planar graphs for other examples of different degrees and diameters.

We have $5n=2m$ since it is 5-regular and every face must be of size 3 or more if there are no multiple edges, so $2m\geq 3f$. Putting these together with Euler's formula we get
$$
\frac{2m}{3} \geq f = 2 + m - n = 2 + m - \frac{2m}{5}
$$
Simplifying we see that $m \geq 30$ and so $n\geq 12$.
If you are allowing loops (not just multiple edges) then the best possible is
10 vertices, achieved by adding loops to the vertices of valency 3 in this graph:

Any graph with more vertices would lead (upon removal of multiple edges or loops) to a simple planar graph of maximum degree at most 5 with diameter 2 but the largest such graph is proven by Yang, Yuansheng, Lin, Jianhua, Dai, Yongjie (J. Comput. Appl. Math. 144 (2002) 349-358) to have 10 vertices, such as the example above.