Question

Suppose that G is a graph with v vertices and e edges and that the degree of each vertex is at least δ, and at most Δ. Show that v⋅δ​/2 ≤ e ≤ v⋅Δ​/2

Answer 1

To prove v⋅δ​/2 ≤ e ≤ v⋅Δ​/2, start by considering the minimum and maximum number of edges that can be present in the graph based on the degrees of the vertices. This will establish the inequality.

Step-by-step explanation:

To prove v⋅δ​/2 ≤ e ≤ v⋅Δ​/2:

1. Start by considering the minimum number of edges that can be present in the graph. If every vertex has a degree of at least δ, then the minimum number of edges is v⋅δ/2, because each edge contributes to 2 vertices.
2. Next, consider the maximum number of edges that can be present in the graph. If every vertex has a degree of at most Δ, then the maximum number of edges is v⋅Δ/2, following the same logic as above.
3. Therefore, we have v⋅δ​/2 ≤ e ≤ v⋅Δ​/2, which proves the inequality.