24.1.8.1. Complete Graph

Now let us define some special types of undirected graphs. So, the first special type of undirected graph is a complete graph. The property of this graph is that you have exactly one edge between each pair of distinct vertices. And since this is a simple graph and our property is that you cannot have more than one edge between every pair of distinct vertices. Automatically we have here the restriction that you cannot have a self-loop, because if you have a self-loop then that self-loop will be violating the definition here.

A complete graph is defined as a graph where each possible pair of distinct vertices is connected by a single edge. If you have n vertices in a complete graph, denoted by K_n, each vertex connects to every other vertex. Since a complete graph must have exactly one edge between any two vertices, it cannot contain self-loops; a self-loop would add an extra edge between a vertex and itself, violating the property of having only one edge between distinct pairs.

Imagine a party where everyone knows each other. Each person in the party is a vertex, and each handshake between two people is an edge. In a complete graph scenario, every person has shaken hands with everyone else exactly once, just like how in a complete graph, every vertex connects to every other. If someone were to shake hands with themselves, it would break the rule of only shaking hands once with every other distinct person at the party.

So, the requirement here is that you take the vertices, all the vertices between every pair of distinct vertices you will have exactly one edge. You do not have the option of 0 or 1, exactly 1 edge should be there between every pair of distinct vertices and if n is the number of nodes in a complete graph then we use this notation K to denote a complete graph with n nodes.

In mathematical terms, if you have n vertices in a complete graph, we denote it as K_n. This notation clearly describes the complete graph's structure where n indicates the total vertices and each pair has one and only one edge connecting them. For example, K_3 represents a complete graph with 3 vertices, which forms a triangle where each vertex is connected to the other two vertices.

Think of a complete graph as a tightly-knit team where each member communicates directly with every other member. If there are 4 members in this team, represented as K_4, each member can directly speak with every other member without needing to go through someone else. This ensures that there's open communication and collaboration, just as there’s a direct edge between every pair of vertices in K_4.

So, these are the some examples of complete graphs with various values of n. The complete graph with 7 nodes is this, complete graph with 4 nodes is this, complete graph with 12 nodes is this.

Complete graphs can exist with different numbers of vertices. For example, K_4 indicates a complete graph with 4 vertices where each of the 4 vertices connects with 3 other vertices, yielding a total of 6 edges. Similarly, K_7 would have 7 vertices and 21 edges because every vertex connects to every other vertex exactly once. Thus, the number of edges in a complete graph with n vertices can be calculated using the formula n(n-1)/2.

Imagine a sports tournament where every player must compete against each other exactly once. If there are 4 players, every player competes against the other 3, forming a complete graph of matches, or K_4. As more players join, the complexity increases; with 7 players, the number of matches rises significantly, mirroring the increase in edges as n grows larger in the complete graph.

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