A wheel $W_n$ is a graph where there are $n$ vertices essentially in a circle, and $1$ vertex in the middle that is adjacent to all the $n$ vertices in the circle. So for example, $W_3$ has $4$ vertices.
How would I find the number of subgraphs with at least one vertex for $W_3$?
Is there a generalized way to find this, i.e. number of subgraphs with at least one vertex for $W_n$?
The n-th wheel has $2^{n+1}$ - 1 not empty subgraphs because
each subgraph is a subset of a set with n + 1 elements.
Exercise. How many not empty, connected
subgraphs does the n-th wheel have?