To find the number of edges in $G$, let's analyze the given conditions and use the properties of connected graphs.

Problem Analysis

1. Graph $G$ is connected: This means there is a path between any pair of vertices.
2. For any non-empty subset $W \subset V$, the subgraph induced by $W$ is also connected: This implies that any subset of vertices in $G$ forms a connected subgraph.

Given these conditions, let's reason about the structure of $G$ and find the number of edges in terms of $n = |V|$.

Key Insight: $G$ Must Be a Complete Graph

If the subgraph induced by any subset of vertices is connected, this requirement is only satisfied by a complete graph:

• A complete graph $K_n$ is one in which every pair of distinct vertices is connected by a unique edge.
• In a complete graph, any subset of vertices remains connected because every vertex is directly connected to every other vertex in the subset.

Therefore, $G$ must be a complete graph $K_n$.

Counting the Edges in a Complete Graph $K_n$

For a complete graph $K_n$ with $n$ vertices:

• Each vertex is connected to every other vertex.
• The total number of edges in $K_n$ is given by the formula: $|E| = \binom{n}{2} = \frac{n(n - 1)}{2}$

Conclusion

The number of edges $|E|$ in $G$ is: $|E| = \frac{n(n - 1)}{2}$

Proof

1. By the given conditions, for any non-empty subset $W \subset V$, the subgraph induced by $W$ is connected.
2. This condition can only hold if every possible subset of vertices is interconnected by edges, which implies that $G$ is a complete graph $K_n$.
3. The number of edges in a complete graph $K_n$ with $n$ vertices is $\frac{n(n - 1)}{2}$, as each pair of vertices has an edge between them.

Thus, we have shown that: $|E| = \frac{n(n - 1)}{2}$

Would you like more details on any part of the proof or further examples?

Here are some additional questions you might find interesting:

1. What would happen if $G$ were not connected initially?
2. How would the number of edges change if we allow disconnected subsets?
3. Is it possible to have a graph where every subset is connected without being complete?
4. How does this result relate to minimal connected graphs like trees?
5. What are the properties of subgraphs induced by non-complete graphs?

Tip: When reasoning about graph properties, always consider whether constraints on connectivity imply the graph is fully connected or if there might be exceptions.