Definition:Adjacent (Graph Theory)/Vertices

Definition

Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph.

Two vertices $u, v \in V$ of $G$ are adjacent if and only if there exists an edge $e = \set {u, v} \in E$ of $G$ to which they are both incident.

Digraph

Let $G = \struct {V, E}$ be a digraph.

Two vertices $u, v \in V$ of $G$ are adjacent if and only if there exists an arc $e = \tuple {u, v} \in E$ of $G$ to which they are both incident.

Non-Adjacent

Let $G = \struct {V, E}$ be a graph.

Two vertices $u, v \in V$ of $G$ are non-adjacent if and only if they are not adjacent.

Also known as

Adjacent elements of a graph can also be described as neighboring (British English spelling: neighbouring).

Also see

• Results about adjacency in the context of graph theory can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): adjacent: 1. a. (of a pair of vertices in a graph)
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 2.3.4.1$: Free Trees
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): adjacent: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): graph: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): adjacent: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): graph: 2.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): adjacent vertices
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): adjacent vertices