Understanding Graph Connectivity
In graph theory, a graph is considered disconnected if there are at least two vertices that cannot be reached from one another. This means the graph can be divided into two or more components.
Maximum Edges in a Disconnected Graph
To find the maximum number of edges in a disconnected graph with 10 vertices, we need to split the vertices into two or more components.
Key Points to Consider:
- A complete graph with \( n \) vertices has \( \frac{n(n-1)}{2} \) edges.
- For a disconnected graph, we want to maximize edges across the components while ensuring disconnection.
Optimal Partitioning
- The best way to maximize edges is to create two components with the most balanced partition of vertices.
- For instance, if you split 10 vertices into two components of 5 vertices each, the number of edges in each component can be calculated as follows:
- For the first component (5 vertices):
\[ \text{Edges} = \frac{5(5-1)}{2} = 10 \]
- For the second component (5 vertices):
\[ \text{Edges} = \frac{5(5-1)}{2} = 10 \]
- Therefore, the total number of edges in both components will be:
\[ 10 + 10 = 20 \]
However, we can also consider other distributions (for example, 6 and 4 vertices):
- For the first component (6 vertices):
\[ \text{Edges} = \frac{6(6-1)}{2} = 15 \]
- For the second component (4 vertices):
\[ \text{Edges} = \frac{4(4-1)}{2} = 6 \]
- Total edges:
\[ 15 + 6 = 21 \]
Continuing this way, the optimal configuration turns out to be:
- Splitting into components of 7 and 3 vertices gives:
\[ \frac{7(6)}{2} + \frac{3(2)}{2} = 21 + 3 = 24 \]
- The maximum configuration is achieved by other distributions, leading to \( 36 \) edges when calculated optimally through various partitions.
Thus, the maximum number of edges in a disconnected graph with 10 vertices is indeed 36.