Complete Graph
🌐 What Is a Complete Graph?
A complete graph is a graph in which:
👉 Every vertex has a direct edge to every other vertex.
If a graph has n vertices, then each vertex connects to n – 1 others.
We represent a complete graph with Kₙ, where “n” is the number of vertices.
- K₂ → 2 vertices, 1 edge
- K₃ → 3 vertices, triangle
- K₄ → 4 vertices, fully connected square
- and so on…
🖼️ Diagrams of Complete Graphs
⭐ Complete Graph K₂
A ----- BOnly two vertices, so one edge is enough.
⭐ Complete Graph K₃
A
/ \
B---CA perfect triangle.
Everyone is connected to everyone.
⭐ Complete Graph K₄
A
/|\
/ | \
B--|--C
\ | /
\|/
DA full “web” of connections among four points.
🧠 How Many Edges Are in a Complete Graph?
If every vertex connects to every other vertex, how do we find the total number of edges?
You can’t just count one-by-one forever, so there’s a nice formula:
[
\text{Edges in } K_n = \frac{n(n-1)}{2}
]
Why divide by 2?
Because if A connects to B, we shouldn’t double-count B connecting back to A — it’s the same edge.
Example
For K₄:
[
\frac{4 \times 3}{2} = 6 \text{ edges}
]
And you’ll see exactly 6 lines in the diagram.
🏡 A Friendly Real-Life Analogy
Think of a WhatsApp group where:
- Everyone messages everyone
- No one needs someone else to forward anything
- All members are directly reachable
That’s a complete graph — no indirect connections, no missing links.
Or imagine a tiny sports team where each player passes the ball to every other teammate without restrictions.
🎯 Why Are Complete Graphs Important?
Complete graphs help us understand:
✔ Maximum possible connections in a network
✔ Worst-case scenarios in algorithms
✔ How dense a graph can become
✔ Benchmark structures in data structures
When you know what “fully connected” looks like, it becomes easier to compare other graphs.
How to Recognize a Complete Graph
When you look at a graph, ask:
- Is every vertex connected to all others?
- Is there any missing edge?
- Does each vertex have degree n – 1?
If all answers are yes, you are looking at a complete graph.
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