Question

Challenge: A complete directed graph is a directed graph whose underlying graph is a complete graph. Show that the sum of the squares of the indegrees over all vertices is equal to the sum of the squares of the outdegrees over all vertices in any directed complete graph.

Short answer

The sum of the squares of indegrees equals the sum of the squares of outdegrees for any directed complete graph.

Step-by-step solution

Understanding the Graph

A complete directed graph means every vertex has a directed edge to every other vertex. If there are \( n \) vertices, each vertex connects to \( n-1 \) other vertices both incoming and outgoing.

Indegree and Outdegree Definitions

For any vertex \( v \), the indegree, \( ext{in}(v) \), is the count of edges directed into \( v \) and is \( n-1 \) since every other vertex connects to it. Similarly, the outdegree, \( ext{out}(v) \), is also \( n-1 \) because \( v \) connects to every other vertex.

Formulating the Problem

For this directed complete graph, the objective is to show: \[ \sum_{v} ext{in}(v)^2 = \sum_{v} ext{out}(v)^2 \]

Calculating Indegree and Outdegree Sums

Calculate the sum of the squares of the indegrees: \[ \sum_{v} ext{in}(v)^2 = n(n-1)^2 \] because there are \( n \) vertices each with an indegree of \( n-1 \). The same applies for the outdegrees: \[ \sum_{v} ext{out}(v)^2 = n(n-1)^2 \].

Concluding the Equality

Since \( \sum_{v} ext{in}(v)^2 = \sum_{v} ext{out}(v)^2 = n(n-1)^2 \), the equality holds. Therefore, the sum of the squares of indegrees is equal to the sum of the squares of outdegrees in a directed complete graph.

Key concepts

Indegree

In the world of graph theory, the term **indegree** is used to describe the number of incoming edges to a specific vertex within a directed graph. Imagine each vertex as a person, and each incoming edge as a message received. The indegree tells you how many messages that person gets from others in the group.

In a complete directed graph—where every vertex is connected to every other vertex by a directed edge—each vertex would have an indegree equal to the number of vertices minus one ( -1). This is because every other vertex in the graph has an edge pointing towards our chosen vertex.

• Example: If there are 5 vertices, each vertex will have an indegree of 4.

Understanding indegree is crucial when analyzing how information may flow into different nodes in network scenarios. By evaluating these flows, one can determine the influence or significance of each vertex in the graph.

Outdegree

The concept of **outdegree** refers to the number of outgoing edges from a specific vertex in a directed graph. To continue with our analogy, imagine each vertex as a person sending messages: the outdegree counts how many messages that person sends to others.

In a complete directed graph, each vertex has an outdegree that is also equal to the number of vertices minus one ( -1). Each vertex sends one edge to all other vertices in the graph, ensuring that each vertex is fully connected outwardly.

• Example: In the same graph of 5 vertices, each vertex will have an outdegree of 4.

This concept helps us to consider the reach or influence of a node since it shows how information or influence can spread from one point to others. By balancing indegree and outdegree, especially in complete graphs, one can ensure that each vertex has optimal connectivity.

Complete Graph

A **complete graph** is a type of simple graph in which there is a unique edge connecting every pair of vertices. When talking about directed graphs, a **directed complete graph** ensures there is a one-way edge (like an arrow) connecting each vertex to all other vertices, creating an interconnected system.

This concept is reflected in how medieval kingdoms might be imagined: each kingdom (vertex) having a direct trade route (edge) to every other kingdom, ensuring that goods and information flow freely and directly between all.

• Features: A directed complete graph maintains balance by having each vertex with equal indegree and outdegree, which are both -1.

The beauty of a complete graph is its symmetry and balance, which provide strong connectivity across the entire structure. This interconnectedness can be incredibly useful in network designs, as it ensures no vertex is isolated, promoting maximum efficiency and robustness within connectivity.