Symmetric Flexible and Rigid Graphs

This module implements functionality for investigating rigidity and flexibility of graphs with a symmetry.

Methods

SymmetricFlexRiGraph

CnSymmetricFlexRiGraph

Unsorted

Cn_symmetries_gens()	Return the list of generators of Cn symmetries of the graph. An element $\omega$ of order n of the automorphism group of the graph generates a $\mathcal{C}_n$-symmetry of the graph if - each partially invariant is invariant - the set of invariant vertices is independent.
cyclic_subgroups()	Return all cyclic subgroups of group with given order.
edge_orbits()	Return the orbits of edges.
invariant_vertices()	Return the invariant vertices.
is_Cn_symmetry()	Return whether sigma generates a $\mathcal{C}_n$-symmetry of the graph.
is_cyclic_subgroup()	Return if a group is cyclic, a generator and order.
set_symmetric_positions()	Given a dictionary of positions of one vertex from some orbits, the other vertices in the orbits are set symmetrically.
vertex_orbits()	Return the orbits of vertices.

AUTHORS:

• Jan Legerský (2020-03-12): initial version

TODO:

• missing documentation of methods
• missing doctests in methods
• finish Cn-symmetry functionality (NACs, doc, classification)

WARNING:

This module is still under development!

SymmetricFlexRiGraph

class flexrilog.symmetric_flexible_rigid_graph.CnSymmetricFlexRiGraph(data, symmetry, pos=None, name=None, check=True)

Bases: flexrilog.symmetric_flexible_rigid_graph.SymmetricFlexRiGraph

This class is inherited from SymmetricFlexRiGraph. It represents a graph with a given $\mathcal{C}_n$ symmetry, namely, a cyclic subgroup of order n of the automorphism group of the graph such that

• each partially invariant is invariant
• the set of invariant vertices is independent.

WARNING:

Only $\mathcal{C}_n$-symmetric NAC-colorings are considered in an instance of CnSymmetricFlexRiGraph for parent methods! For example, FlexRiGraph.NAC_colorings() returns the list of all $\mathcal{C}_n$-symmetric NAC-colorings of the graph.

INPUT:

• data: provides the information about edges, see FlexRiGraph..
• symmetry – sage.graphs.graph.Graph that is a subgroup of the automorphism group of the graph or the list of its generator. The properties above must hold.

TODO:

• examples
• check input as list of edges
• input only generator

static Cn_symmetries_gens(graph, n)

Return the list of generators of Cn symmetries of the graph.

An element $\omega$ of order n of the automorphism group of the graph generates a $\mathcal{C}_n$-symmetry of the graph if

• each partially invariant is invariant
• the set of invariant vertices is independent.

NAC_colorings()

Return $\mathcal{C}_n$-symmetric NAC-colorings.

static cyclic_subgroups(group, order)

Return all cyclic subgroups of group with given order.

edge_orbits()

Return the orbits of edges.

invariant_vertices()

Return the invariant vertices.

static is_Cn_symmetry(graph, sigma, n)

Return whether sigma generates a $\mathcal{C}_n$-symmetry of the graph.

static is_cyclic_subgroup(subgroup)

Return if a group is cyclic, a generator and order.

set_symmetric_positions(pos)

Given a dictionary of positions of one vertex from some orbits, the other vertices in the orbits are set symmetrically.

vertex_orbits()

Return the orbits of vertices.

class flexrilog.symmetric_flexible_rigid_graph.SymmetricFlexRiGraph(data, symmetry, pos=None, name=None, check=True)

Bases: flexrilog.flexible_rigid_graph.FlexRiGraph

The class SymmetricFlexRiGraph is inherited from FlexRiGraph. It represents a graph with a given symmetry.

INPUT:

• data: provides the information about edges, see FlexRiGraph..
• symmetry – sage.graphs.graph.Graph that is a subgroup of the automorphism group of the graph or the list of its generators

TODO:

examples