Degree (graph theory)

The degree of a vertex is the number of incident edge ends, so a loop contributes two in an undirected graph. In a directed graph, in-degree and out-degree count incoming and outgoing edges separately.

Graph (discrete mathematics): A structure consisting of vertices and edges that connect pairs of vertices. Degree measures how many edge ends meet each vertex in this structure.
Handshaking lemma: The sum of all vertex degrees in a finite undirected graph equals twice its number of edges. It follows by counting each edge through its two ends.
Vertex connectivity: The minimum number of vertices whose removal disconnects a graph or reduces it to one vertex. Minimum degree constrains how many vertices must be removed to disconnect a graph.
Neighborhood (graph theory): The neighborhood of a vertex is the set of vertices adjacent to it. Its size can differ from degree when loops or parallel edges occur.
Vertex (graph theory): A vertex is an element of a graph, often represented as a point. Degree is assigned to each vertex individually.
Degree sequence: A list of the degrees of all vertices in a graph, usually arranged in nonincreasing order. It records every local degree as a graph-wide signature.
Erdős–Gallai theorem: A criterion determining whether a finite nonincreasing sequence of nonnegative integers is a simple graph's degree sequence. It tests whether proposed local degrees can belong to one simple graph.
Adjacency matrix: A matrix encoding which pairs of vertices in a graph are adjacent. For simple graphs, row sums give vertex degrees.
Edge (graph theory): An edge is a connection between vertices in a graph. Counting the edges incident to a vertex defines its degree.
Minimum degree: The smallest vertex degree in a graph, commonly denoted δ. It gives a lower bound on every vertex's number of incident edges.

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