To convert a graph into a simplicial complex, first identify all complete subgraphs (cliques) within the graph. Note that these cliques can overlap but do not collectively make a larger complete graph unless fully connected.
Conditions: The starting structure is an arbitrary graph.; The graph contains complete subgraphs (cliques).
To convert a graph into a simplicial complex, first identify all complete subgraphs (cliques) within the graph. Note that these cliques can overlap but do not collectively make a larger complete graph unless fully connected.
Conditions: The starting structure is an arbitrary graph.; The graph contains complete subgraphs (cliques).
You interpret a complete graph as a simplex by keeping its vertices and edges, then filling in higher-dimensional cells such as 2-dimensional triangular faces, 3-dimensional solids, and 4-dimensional spaces. A complete graph on k vertices forms a(k−1)-simplex.
Conditions: The starting structure is a complete graph on k vertices.; Higher-dimensional cells are filled successively between existing lower-dimensional ones.
You interpret a complete graph as a simplex by keeping its vertices and edges, then filling in higher-dimensional cells such as 2-dimensional triangular faces, 3-dimensional solids, and 4-dimensional spaces. A complete graph on k vertices forms a(k−1)-simplex.
Conditions: The starting structure is a complete graph on k vertices.; Higher-dimensional cells are filled successively between existing lower-dimensional ones.
Individual simplices in a simplicial complex can be completely separate or stuck together by sharing a face, an edge, or a vertex. Crucially, the intersection must happen along an entire face, never just a partial face.
Conditions: The object is a simplicial complex.; Intersections occur between any two simplices within the complex.
Individual simplices in a simplicial complex can be completely separate or stuck together by sharing a face, an edge, or a vertex. Crucially, the intersection must happen along an entire face, never just a partial face.
Conditions: The object is a simplicial complex.; Intersections occur between any two simplices within the complex.