Adjacency

Adjacency refers to the relation of being immediately next to or directly connected. In mathematics and related disciplines, the notion is used to describe how objects are positioned relative to each other within a given structure, such as a graph, a map, or a geometric space.

In graph theory, two vertices are adjacent if they share an edge. This adjacency relation is symmetric in undirected graphs and may be oriented in directed graphs. For a vertex v, the set of adjacent vertices is called its neighborhood. Graphs can be represented by an adjacency matrix, where the entry A[i][j] records whether i and j are adjacent, and by an adjacency list, which lists the neighbors of each vertex.

In planar maps and geometry, regions or objects are adjacent if they share a common boundary segment or point, with typical emphasis on a nontrivial boundary. In data structures and geographic information systems, adjacency information supports spatial queries and neighborhood-based algorithms. In social networks and other relational systems, adjacency describes direct connections between entities.

Adjacency is foundational to algorithms for traversal, path finding, clustering, and network analysis. It also relates to related concepts such as incidence, neighborhood, and degree, which measure how many adjacencies a given object has.

See also: adjacency matrix, adjacency list, graph theory, neighborhood.