# Complex

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

## Definition

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

### Edge colored graph

The flag graph of a tetrahedron is a complex.

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 ${\displaystyle A(i)}$ 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 ${\displaystyle i \neq j}$ then the intersection ${\displaystyle A(i)\cup A(j)}$ is empty.
• The graph ${\displaystyle (V,A)}$ is connected.

### Permutation sequence

An n-complex is a set ${\displaystyle \Omega}$ and a sequence of permutations on ${\displaystyle \Omega}$, ${\displaystyle A : \{1,\dots,n\} \rightarrow \mathrm{Aut}(\Omega)}$ such that:

• For every ${\displaystyle A(i)}$ in A, ${\displaystyle A(i)\circ A(i)=\mathrm{id}}$
• For every ${\displaystyle A(i)}$ in A, there is no element of ${\displaystyle \Omega}$ which is a fixed point of ${\displaystyle A(i)}$.
• If ${\displaystyle i \neq j}$ then for any element ${\displaystyle x\in\Omega}$, ${\displaystyle A(i)(x)\neq A(j)(x)}$.
• The group generated by the permutations in A is transitive on ${\displaystyle \Omega}$.

## Concepts

### Elements

The edges, faces and vertices of tetrahedron flag graph shown above.

For a given i the elements of a complex of rank i correspond to the orbits of the group generated by permutations ${\displaystyle A(j)}$ where ${\displaystyle i\neq j}$.[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

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

## Bibliography

• 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.