Degree (graph theory)
Introduction
In the field of graph theory, a branch of mathematics concerned with the study of graphs, the concept of a "degree" is fundamental. The degree of a vertex in a graph is the number of edges incident to the vertex, with loops counted twice. This concept is pivotal in understanding the structure and properties of graphs, influencing various algorithms and applications across computer science, network analysis, and combinatorics.
Definitions and Basic Concepts
The degree of a vertex \( v \) in a graph \( G \), denoted as \( \deg(v) \), is defined as the number of edges connected to \( v \). In a simple graph, which contains no loops or multiple edges, the degree is simply the count of adjacent vertices. However, in a multigraph, where multiple edges between vertices are allowed, each edge contributes to the degree count. In the case of a loop, it contributes two to the degree of the vertex it is incident to.
For a directed graph, also known as a digraph, the concept of degree is split into two: the indegree and the outdegree. The indegree of a vertex is the number of incoming edges, while the outdegree is the number of outgoing edges. The total degree of a vertex in a digraph is the sum of its indegree and outdegree.
Properties of Vertex Degrees
One of the fundamental properties of vertex degrees in any graph is the Handshaking Lemma, which states that the sum of the degrees of all vertices in a graph is twice the number of edges. Mathematically, for a graph \( G = (V, E) \) with vertex set \( V \) and edge set \( E \), this is expressed as:
\[ \sum_{v \in V} \deg(v) = 2|E| \]
This lemma is a direct consequence of the fact that each edge contributes exactly two to the total degree count, one for each of its endpoints.
In a regular graph, every vertex has the same degree. If each vertex has degree \( k \), the graph is called \( k \)-regular. Regular graphs are of particular interest due to their symmetrical properties and applications in network design.
Degree Sequences
The degree sequence of a graph is a list of its vertex degrees, usually written in non-increasing order. For example, a graph with vertex degrees 3, 3, 2, 2, 2 has a degree sequence of (3, 3, 2, 2, 2). Degree sequences are useful in characterizing certain classes of graphs and are central to the study of graph realization problems, where one seeks to determine whether a given sequence can be represented as the degree sequence of some graph.
A classic result related to degree sequences is the Havel-Hakimi algorithm, which provides a constructive method to determine if a sequence of non-negative integers can be the degree sequence of a simple graph.
Degree Centrality in Network Analysis
In the context of network analysis, degree centrality is a measure of the importance or influence of a vertex within a network. It is defined as the degree of the vertex normalized by the maximum possible degree in the network. High degree centrality indicates a vertex with many connections, often interpreted as a hub or influential node within the network.
Degree centrality is particularly useful in social network analysis, where it helps identify key individuals or entities within a social structure. It is also applied in biological networks, communication networks, and the analysis of internet topology.
Applications and Algorithms
The concept of vertex degree is integral to many algorithms in graph theory. For instance, in graph coloring, the degree of vertices can influence the choice of coloring strategy. In graph traversal algorithms like Breadth-First Search and Depth-First Search, the degree of vertices affects the traversal order and efficiency.
In network design, ensuring certain degree constraints can lead to more robust and efficient networks. For example, in designing a communication network, maintaining a minimum degree for each node can ensure redundancy and fault tolerance.