Degree (graph theory)

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge.^[1] The degree of a vertex ${\displaystyle v}$ is denoted ${\displaystyle \deg(v)}$ or ${\displaystyle \deg v}$. The maximum degree of a graph ${\displaystyle G}$ is denoted by ${\displaystyle \Delta (G)}$, and is the maximum of ${\displaystyle G}$'s vertices' degrees. The minimum degree of a graph is denoted by ${\displaystyle \delta (G)}$, and is the minimum of ${\displaystyle G}$'s vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0.

In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted ${\displaystyle K_{n}}$, where ${\displaystyle n}$ is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, ${\displaystyle n-1}$.

In a signed graph, the number of positive edges connected to the vertex ${\displaystyle v}$ is called positive deg${\displaystyle (v)}$ and the number of connected negative edges is entitled negative deg${\displaystyle (v)}$.^[2][3]