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Chapter 1: Problem 1

What is the number of edges in a \(K^{n}\) ?

Short Answer

Expert verified
The number of edges in a \(K^{n}\) is given by the formula \(n(n-1)/2\), where \(n\) is the number of vertices of the graph.

Step by step solution

01

Understand the problem

The problem is asking for the number of edges in a complete \(K^{n}\) graph which is a graph where each pair of graph vertices is connected by an edge.
02

Applying the formula

The formula for calculating the number of edges in a complete graph is \(n(n-1)/2\). This represents the maximum number of edges such a graph can have, which happens when each node is connected to every other node.
03

Calculation

With \(n\) vertices, the number of possible edges is given by the formula. Substitute \(n\) into the formula \(n(n-1)/2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Theory
Graph theory is a branch of mathematics that explores the relationships between pairs of objects. These objects, often called vertices or nodes, are connected by links known as edges. Graph theory helps in understanding complex networks by simplifying them into structures called graphs. Each graph is made up of vertices connected by edges, and they can represent anything from social networks and communication systems to computer networks.
  • Vertex or Node: The individual entities in a graph.
  • Edge: The connection between two vertices.
  • Complete Graph: A type of graph where every vertex is connected to every other vertex.
In a complete graph like the complete \(K^n\), each vertex is directly connected with every other vertex, making it a perfectly connected structure.
Edges in Graphs
Edges are the backbone of any graph. They create pathways between vertices and define the structure's complexity. In a graph, especially in a complete graph, the number of edges is pivotal as it indicates how interconnected the vertices are.
  • Direct Connections: In complete graphs, each vertex connects directly to all other vertices.
  • Graph Density: This relates to how many edges there are compared to the number of possible edges.
For a complete graph \(K^n\), there is a consistent pattern in the number of edges. The more vertices in the graph, the more edges you can expect, as every vertex pairs with every other vertex. This aspect is often critical for understanding networks that aim for maximum connectivity, such as telecommunications networks.
Mathematical Formula
The mathematical formula for finding the number of edges in a complete graph \(K^n\) is \( \frac{n(n-1)}{2} \). Let's take a closer look at why this formula works. The formula calculates the total number of unique edge connections that can be formed between \(n\) vertices.
  • Combination Concept: The formula is derived from the combination of \(n\) items taken two at a time \( \binom{n}{2} \) because each edge connects two vertices.
  • Simplification: Since the order of connection doesn鈥檛 matter, each pair is only counted once.
This formula simplifies the process of counting edges in large graphs, making it efficient to determine how interconnected a graph can be. When you plug in the value of \(n\) into \(n(n-1)/2\), you get the exact count of edges, essential for analyzing graph properties.

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Most popular questions from this chapter

Let \(d \in \mathbb{N}\) and \(V:=\\{0,1\\}^{d} ;\) thus, \(V\) is the set of all \(0-1\) sequences of length \(d .\) The graph on \(V\) in which two such sequences form an edge if and only if they differ in exactly one position is called the \(d\)-dimensional cube. Determine the average degree, number of edges, diameter, girth and circumference of this graph. (Hint for the circumference: induction on \(d\).)

Show that a graph is bipartite if and only if every induced cycle has even length.

Is there a function \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that, for all \(k \in \mathbb{N}\), every graph of minimum degree at least \(f(k)\) is \(k\)-connected?

\({ }^{+}\)Find a good lower bound for the order of a connected graph in terms of its diameter and minimum degree.

Prove that the cycles and the cuts in a graph together generate its entire edge space, or find a counterexample.
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