Link (graph theory)
Encyclopedia
In geometry
, the link of a vertex
of a 2-dimension
al simplicial complex
is a graph
that encodes information about the local structure of the complex at the vertex.
It is a graph-theoretic analog to a sphere centered at a point.
The link of a vertex of a tetrahedron is a triangle – the three vertices of the link corresponds to the three edges incident to the vertex, and the three edges of the link correspond to the faces incident to the vertex. In this example, the link can be visualized by cutting off the vertex with a plane; formally, intersecting the tetrahedron with a plane near the vertex – the resulting cross-section is the link.
be a simplicial complex. The link 
of a vertex 
of 
is the graph constructed as follows. The vertices of 
correspond to edges of 
which are incident to 
. Two such edges are adjacent
in
if they are incident
to a common 2-cells at
. In general, for an abstract simplicial complex
and a face
of 
, denoted 
is the set of faces 
such that G 
F = 
and G 
F 
X. Because 
is simplicial, there is a set isomorphism between 
and 
such that F 
.
The graph
is often given the topology
of a ball
of small radius centred at
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the link of a vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
of a 2-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
is a graph
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
that encodes information about the local structure of the complex at the vertex.
It is a graph-theoretic analog to a sphere centered at a point.
Example
Definition
Let be a simplicial complex. The link of a vertex of is the graph constructed as follows. The vertices of correspond to edges of which are incident to . Two such edges are adjacentAdjacent
Adjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
in
if they are incidentIncidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
to a common 2-cells at
. In general, for an abstract simplicial complexAbstract simplicial complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
and a face
of , denoted is the set of faces such that G F = and G F X. Because is simplicial, there is a set isomorphism between and such that F .The graph
is often given the topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of a ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
of small radius centred at
.