Find The Degree Of Each Vertex In The Graph Calculator

Calculate Vertex Degrees

What is the Degree of Each Vertex in a Graph?

In graph theory, the degree of each vertex in a graph is a fundamental concept. For a given vertex (or node) in a graph, its degree is the number of edges that are incident to it (connected to it). In simpler terms, it’s the number of direct connections a vertex has to other vertices in the graph.

For undirected graphs, the degree of a vertex `v`, denoted as `deg(v)`, is simply the count of edges connected to `v`. Loops (edges connecting a vertex to itself) are usually counted twice.

For directed graphs, we distinguish between:

• In-degree (deg^–(v)): The number of edges coming *into* vertex `v`.
• Out-degree (deg⁺(v)): The number of edges going *out* from vertex `v`.

The total degree of a vertex in a directed graph is the sum of its in-degree and out-degree.

Understanding the degree of each vertex in a graph is crucial for analyzing the graph’s structure, identifying important nodes (like hubs), and in various algorithms related to graph traversal and network analysis.

Who should use it? Students learning graph theory, computer scientists, network analysts, data scientists, and anyone working with graph-based data structures or network models will find the degree of each vertex in a graph a vital metric.

Common misconceptions: A common mistake is not differentiating between directed and undirected graphs when calculating degrees, or miscounting loops in undirected graphs.

Degree of Each Vertex in a Graph Formula and Mathematical Explanation

The calculation of the degree of each vertex in a graph depends on whether the graph is directed or undirected, and how it’s represented (e.g., adjacency matrix or adjacency list).

Undirected Graphs

For an undirected graph `G = (V, E)`, where `V` is the set of vertices and `E` is the set of edges, the degree of a vertex `v ∈ V`, `deg(v)`, is the number of edges in `E` that have `v` as an endpoint. If the graph is represented by an adjacency list, `deg(v)` is the number of neighbors listed for `v`. Loops are counted twice.

The Handshaking Lemma states that the sum of the degrees of all vertices in an undirected graph is equal to twice the number of edges: `∑ deg(v) = 2|E|`.

Directed Graphs

For a directed graph `G = (V, E)`, for a vertex `v ∈ V`:

• In-degree (deg^–(v)): The number of edges `(u, v) ∈ E` (edges pointing *to* `v`).
• Out-degree (deg⁺(v)): The number of edges `(v, u) ∈ E` (edges pointing *from* `v`).

In a directed graph, the sum of all in-degrees equals the sum of all out-degrees, which equals the number of edges: `∑ deg<sup>–</sup>(v) = ∑ deg<sup>+</sup>(v) = |E|`.

Variables Table

Variable	Meaning	Unit	Typical Range
V	Set of vertices (nodes)	–	Finite set
E	Set of edges (links)	–	Finite set of pairs (undirected) or ordered pairs (directed) of vertices
deg(v)	Degree of vertex v (undirected)	Count	0 to |V|-1 (or more with loops/multigraphs)
deg^–(v)	In-degree of vertex v (directed)	Count	0 to |V|-1
deg^+(v)	Out-degree of vertex v (directed)	Count	0 to |V|-1
|V|	Number of vertices	Count	1 to ∞
|E|	Number of edges	Count	0 to |V|(|V|-1) or |V|(|V|-1)/2

Variables used in calculating the degree of each vertex in a graph.

Practical Examples (Real-World Use Cases)

Example 1: Undirected Social Network

Consider a small social network represented as an undirected graph with 4 people (vertices 0, 1, 2, 3) and friendships (edges):

• 0 is friends with 1 and 2
• 1 is friends with 0 and 2
• 2 is friends with 0, 1, and 3
• 3 is friends with 2

Adjacency List Input:

0: 1 2
1: 0 2
2: 0 1 3
3: 2
                

Using the calculator with “Undirected” type:

• deg(0) = 2
• deg(1) = 2
• deg(2) = 3
• deg(3) = 1

Vertex 2 has the highest degree, meaning person 2 is the most connected in this small network.

Example 2: Directed Web Page Links

Consider a small website with 3 pages (0, 1, 2) and links between them (directed edges):

• 0 links to 1
• 1 links to 0 and 2
• 2 links to 0

Adjacency List Input:

0: 1
1: 0 2
2: 0
                

Using the calculator with “Directed” type:

• Vertex 0: In-degree = 2 (from 1, 2), Out-degree = 1 (to 1)
• Vertex 1: In-degree = 1 (from 0), Out-degree = 2 (to 0, 2)
• Vertex 2: In-degree = 1 (from 1), Out-degree = 1 (to 0)

Page 0 receives the most links (in-degree 2), while page 1 links out to the most pages (out-degree 2).

How to Use This Degree of Each Vertex in a Graph Calculator

1. Enter Number of Vertices: Input the total number of vertices in your graph. Vertices are assumed to be numbered from 0 up to this number minus 1.
2. Select Graph Type: Choose ‘Undirected’ or ‘Directed’ based on your graph.
3. Enter Adjacency List: In the text area, provide the graph’s structure as an adjacency list. Each line should represent a vertex and its neighbors (for undirected) or the vertices it points to (for directed). Format: `vertex_number: neighbor1 neighbor2 …`. Ensure vertex numbers are within the range 0 to (Number of Vertices – 1).
4. Calculate: Click the “Calculate Degrees” button or simply make changes to the inputs.
5. View Results:
   • The “Primary Result” section will summarize the degrees found.
   • “Intermediate Values” show the number of edges and graph type.
   • The “Table of Vertex Degrees” lists the degree(s) for each vertex.
   • The bar chart visually represents the degrees.
6. Reset: Click “Reset” to clear inputs and results to default values.
7. Copy Results: Click “Copy Results” to copy the main findings to your clipboard.

Understanding the degree of each vertex in a graph helps identify nodes with many connections, which can be critical points in a network.

Key Factors That Affect Degree of Each Vertex in a Graph Results

• Graph Density: Denser graphs (more edges) tend to have vertices with higher average degrees.
• Graph Type (Directed vs. Undirected): This determines whether you calculate a single degree or separate in-degrees and out-degrees. The interpretation of the degree of each vertex in a graph changes significantly.
• Presence of Loops: In undirected graphs, loops (edges from a vertex to itself) typically add 2 to the degree of that vertex. Our calculator counts each connection listed.
• Presence of Parallel Edges (Multigraphs): If there are multiple edges between the same pair of vertices, each will contribute to the degree. The adjacency list should reflect these multiple connections if they exist (e.g., `0: 1 1`).
• Connectivity of the Graph: A highly connected graph will generally have higher degrees compared to a sparse or disconnected graph.
• Graph Structure: Specific structures like star graphs (one central node), complete graphs (all nodes connected), or line graphs will have very predictable degree distributions. Finding the degree of each vertex in a graph reveals this structure.

Frequently Asked Questions (FAQ)

What is the degree of an isolated vertex?

An isolated vertex (not connected to any other vertex) has a degree of 0 (in-degree=0 and out-degree=0 if directed).

Can the degree of a vertex be negative?

No, the degree (or in/out-degree) is a count of edges and is always non-negative.

What does a high degree indicate for a vertex in a social network?

In a social network, a vertex with a high degree (or high in-degree in a directed network representing ‘follows’) often represents a popular or influential individual/entity.

What is the sum of degrees in any graph?

In an undirected graph, the sum of degrees is twice the number of edges. In a directed graph, the sum of in-degrees equals the sum of out-degrees, which equals the number of edges.

How does the adjacency list format work?

Each line starts with a vertex number, followed by a colon, then a space-separated list of vertices it connects to. For directed graphs, `u: v` means an edge from `u` to `v`.

What if my graph has more vertices than I specified?

The calculator will try to parse based on the adjacency list, but it’s best to ensure the “Number of Vertices” matches the highest vertex index mentioned + 1.

How are loops handled in this calculator?

If you list a vertex as its own neighbor (e.g., `0: 0 1`), it will contribute to its degree count based on the graph type, as per the adjacency list definition.

Why is understanding the degree of each vertex in a graph important for network analysis?

The distribution of degrees can reveal a lot about the network’s topology, robustness, and the roles of different nodes within it. It’s a basic but crucial metric for any .