Calculate Graph Vertex Degrees From Number of Vertices and Edges

What is Vertex Degree?

A vertex (or node) in a graph is a fundamental unit that can represent anything from a person in a social network to a city in a transportation map. The degree of a vertex is simply the count of edges connected to it.

In undirected graphs, the degree is straightforward - it's the number of edges attached to the vertex. In directed graphs, you have both in-degree (edges coming into the vertex) and out-degree (edges going out of the vertex).

How to Calculate Vertex Degrees

Calculating vertex degrees from the total number of vertices and edges involves understanding the Handshaking Lemma, which states that the sum of all vertex degrees in a graph is equal to twice the number of edges.

The basic steps are:

1. Count the total number of vertices (V) in your graph
2. Count the total number of edges (E) in your graph
3. Calculate the sum of all vertex degrees (Σdeg(v)) using the formula Σdeg(v) = 2E
4. Distribute the total degree sum among individual vertices based on your graph's structure

For regular graphs where all vertices have the same degree, the degree of each vertex is simply 2E/V.

The Formula

The fundamental relationship between vertices, edges, and degrees is given by the Handshaking Lemma:

Where:

• Σdeg(v) is the sum of degrees of all vertices
• E is the number of edges in the graph

For a regular graph where each vertex has the same degree:

Where V is the number of vertices.

Worked Example

Let's calculate vertex degrees for a simple graph with 4 vertices and 4 edges.

1. Number of vertices (V) = 4
2. Number of edges (E) = 4
3. Calculate sum of degrees: Σdeg(v) = 2 × 4 = 8
4. Assuming a regular graph, each vertex degree = 8 / 4 = 2

This means each of the 4 vertices in this graph has exactly 2 edges connected to it.