# Complex

A **complex** is an edge-colored graph that generalizes the graph encoding of a map and the flag graph of a polytope.

## Definition[edit | edit source]

There exist several equivalent ways to define the idea of a complex.^{[1]}

### Edge colored graph[edit | edit source]

A complex is a connected properly edge colored simple graph.

To put it another way, a complex is a vertex set V, a color set C, and a function E from colors where yields the edges with color i such that:

- For every color i the every vertex is incident on exactly 1 edge of color i.
- If then the intersection is empty.
- The graph is connected.

### Permutation sequence[edit | edit source]

An n-complex is a set and a sequence of permutations on , such that:

- For every in A,
- For every in A, there is no element of which is a fixed point of .
- If then for any element , .
- The group generated by the permutations in A is transitive on .

## Concepts[edit | edit source]

### Elements[edit | edit source]

For a given i the elements of a complex of rank i correspond to the orbits of the group generated by permutations where .^{[1]}

Alternatively the elements of rank i in a a complex correspond to the connected components in the graph with edges of color i removed.^{[1]}

### Incidence[edit | edit source]

Two elements are incident on each other if their connected components share a vertex.^{[2]}

## References[edit | edit source]

- ↑
^{1.0}^{1.1}^{1.2}Wilson (2012) - ↑ Garza-Vargas & Hubard (2018:7)

## Bibliography[edit | edit source]

- Garza-Vargas, Jorge; Hubard, Isabel (7 July 2018). "Polytopality of Maniplexes" (PDF). arXiv:1604.01164.
- Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators".
*Symmetry*. doi:10.3390/sym4020265.

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