A simple undirected graph has no self-loops and no parallel edges.
Determine how many of the simple undirected graphs $G$ = $(V,E)$ with $V$ = ${1,...,5}$ contain a vertex $v$ of degree $deg(v)$ = $4$
How to find the required number of graphs with V =5.
I could only find the total number of graphs. Couldn't proceed.
Draw the graph with the fewest number of edges: Just a star. A vertex connected to all other ones. Then without breaking the structure we can construct all possible graphs on the other 4 vertices, and there are gonna be exactly $2^{C^{4}_{2}}$ of these. Therefore there will be $2^{C^{4}_{2}}=2^6=64$ graphs.