The problem asks how many subgraphs of $K^-_n$ (a complete graph on $n$ vertices with one edge deleted) are isomorphic to $K^-_5$ (a complete graph on 5 vertices with one edge deleted).

Step 1: Understand the structures

• $K_n$ is a complete graph with $\binom{n}{2}$ edges, and $K^-_n$ is a graph obtained by removing exactly one edge from $K_n$, leaving $\binom{n}{2} - 1$ edges.
• $K_5$ is a complete graph on 5 vertices with $\binom{5}{2} = 10$ edges, and $K^-_5$ has 9 edges (since one edge is removed).

Step 2: Subgraph isomorphism to $K^-_5$

We need to count how many subgraphs of $K^-_n$ are isomorphic to $K^-_5$, which involves choosing 5 vertices and ensuring that the subgraph induced on these 5 vertices is $K^-_5$.

Step 3: Choose 5 vertices

First, we can select 5 vertices from the $n$ vertices of $K^-_n$ in $\binom{n}{5}$ ways.

Step 4: Ensure the subgraph is $K^-_5$

For any set of 5 vertices in $K^-_n$, the subgraph induced on these 5 vertices will either be $K_5$ or $K^-_5$. To have a subgraph isomorphic to $K^-_5$, the deleted edge in $K^-_n$ must lie between these 5 vertices.

The number of ways to choose 5 vertices where one specific edge is missing is the number of ways to pick 5 vertices that include the missing edge. The missing edge can be part of $\binom{5}{2} = 10$ possible edges in a $K_5$, and in $K^-_5$, exactly one of these 10 edges is removed.

Thus, if the deleted edge in $K^-_n$ lies within any of the chosen 5 vertices, the subgraph induced on these vertices will be isomorphic to $K^-_5$.

Step 5: Conclusion

Out of the $\binom{n}{5}$ ways to choose 5 vertices, the number of valid choices where the deleted edge in $K^-_n$ lies within those 5 vertices is exactly $\binom{n}{5}$.

Thus, the number of subgraphs of $K^-_n$ isomorphic to $K^-_5$ is:

$\boxed{\binom{n}{5}}$

Additional Questions:

1. How many edges are in $K_n$ if $n = 6$?
2. What is the difference between $K^-_n$ and $K_n$ in terms of edge count?
3. How would the answer change if more than one edge is deleted from $K_n$?
4. What are the properties of a subgraph isomorphic to $K_4$?
5. How many edges must be deleted from $K_5$ to get a tree?

Tip: Understanding subgraph isomorphisms can help in counting structures in larger graphs, which is a key concept in combinatorics and graph theory.