Neighborhood (graph theory)

In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the induced subgraph of G consisting of all vertices adjacent to v. For example, the image shows a graph of 6 vertices and 7 edges. Vertex 5 is adjacent to vertices 1, 2, and 4 but it is not adjacent to 3 and 6. The neighbourhood of vertex 5 is the graph with three vertices, 1, 2, and 4, and one edge connecting vertices 1 and 2.

The neighbourhood is often denoted N_G(v) or (when the graph is unambiguous) N(v). The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include v itself, and is more specifically the open neighbourhood of v; it is also possible to define a neighbourhood in which v itself is included, called the closed neighbourhood and denoted by N_G[v]. When stated without any qualification, a neighbourhood is assumed to be open.

Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also used in the clustering coefficient of a graph, which is a measure of the average density of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of their neighbourhoods, or by symmetries that relate neighbourhoods to each other.

An isolated vertex has no adjacent vertices. The degree of a vertex is equal to the number of adjacent vertices. A special case is a loop that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood.

If all vertices in G have neighbourhoods that are isomorphic to the same graph H, G is said to be locally H, and if all vertices in G have neighbourhoods that belong to some graph family F, G is said to be locally F (Hell 1978, Sedlacek 1983). For instance, in the octahedron graph shown in the figure, each vertex has a neighbourhood isomorphic to a cycle of four vertices, so the octahedron is locally C₄.

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