In this Chapter we give functions for accessing the cubic edge-transitive graphs stored in this package, and related properties. Currently, this package contains all connected cubic edge-transitive graphs on up to 10000 vertices. For information and references on these graphs, see [CD02] and [CMMP06] (note: these references do not cover the enumeration of graphs on more than 768 vertices, but the authors are currently writing up the extended computations for publishing).
Let Γ be a simple graph (undirected, loopless, without multiple edges). Then Γ is cubic (or trivalent, or 3-valent) if each vertex of the graph has exactly 3 neighbours.
The graph Γ is edge-transitive if the automorphism group of Γ acts transitively on the edges of Γ.
A regular edge-transitive graph can be either vertex-transitive or non-vertex-transitive. In the former case the graph is arc-transitive, and in the latter case it is said to be semisymmetric. Currently in the GraphSym package, if a user wants access to a cubic edge-transitive graph they must also specify if the graph is arc-transitive or semisymmetric.
In this Section we introduce functions for the access to the cubic edge-transitive graphs stored in the GraphSym package.
‣ CubicATGraph( n, i[, data] ) | ( function ) |
Returns: A digraph.
Given positive integers n,i, this function returns the ith cubic arc-transitive graph with n vertices available in this package. If there is no such graph, the function returns fail.
When the optional argument data is specified, it must have value true or false. If data=true, the graph returned by this function will have been assigned the precomputed properties and attributes from SetCubicATGraphProps (3.2-6). If data=false or not specified, none of these properties or attributes are given to the resulting graph.
gap> CubicATGraph(2048,4);
<immutable symmetric digraph with 2048 vertices, 6144 edges>
gap> CubicATGraph(2048,26);
fail
‣ CubicSSGraph( n, i[, data] ) | ( function ) |
Returns: A digraph.
Given positive integers n,i, this function returns the ith cubic semisymmetric graph with n vertices available in this package. If there is no such graph, the function returns fail.
When the optional argument data is specified, it must have value true or false. If data=true, the graph returned by this function will have been assigned the precomputed properties and attributes from SetCubicSSGraphProps (3.2-7). If data=false or not specified, none of these properties or attributes are given to the resulting graph.
gap> CubicSSGraph(4374,10);
<immutable symmetric digraph with 4374 vertices, 13122 edges>
gap> CubicSSGraph(4374,30);
fail
‣ AllCubicATGraphs( n[, data] ) | ( function ) |
Returns: A list.
Given a positive integer n, this function returns a list containing all cubic arc-transitive graphs with n vertices available in this package. If there are no such graphs, the function returns fail.
When the optional argument data is specified, it must have value true or false. If data=true, the graphs returned by this function will have been assigned the precomputed properties and attributes from SetCubicATGraphProps (3.2-6). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> gammas:=AllCubicATGraphs(3072,true);;
gap> Length(gammas);
21
‣ AllCubicSSGraphs( n[, data] ) | ( function ) |
Returns: A list.
Given a positive integer n, this function returns a list containing all cubic semisymmetric graphs with n vertices available in this package. If there are no such graphs, the function returns fail.
When the optional argument data is specified, it must have value true or false. If data=true, the graphs returned by this function will have been assigned the precomputed properties and attributes from SetCubicSSGraphProps (3.2-7). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> gammas:=AllCubicSSGraphs(7680,true);;
gap> Length(gammas);
21
‣ CubicATGraphIterator( n[, data] ) | ( function ) |
Returns: A list
Given a positive integer n, this function returns an iterator over all cubic arc-transitive graphs with n vertices available in this package. If there are such no graphs, the function returns an empty iterator.
When the optional argument data is specified, it must have value true or false. If data=true, the graphs returned by this function will have been assigned the precomputed properties and attributes from SetCubicATGraphProps (3.2-6). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
‣ CubicSSGraphIterator( n[, data] ) | ( function ) |
Returns: A list
Given a positive integer n, this function returns an iterator over all cubic semisymmetric graphs with n vertices available in this package. If there are such no graphs, the function returns an empty iterator.
When the optional argument data is specified, it must have value true or false. If data=true, the graphs returned by this function will have been assigned the precomputed properties and attributes from SetCubicSSGraphProps (3.2-7). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
In this Section we give the functions which give information about the cubic edge-transitive graphs library, and the properties and attributes of the graphs it contains.
Currently, there are no precomputed attributes available for the cubic edge-transitive graphs stored in this package. In the near future, we plan to provide several attributes for each of the cubic edge-transitive graphs stored, including diameter, girth and bipartiteness.
Now we introduce functions which are used to find information about the library and each of the graphs it stores.
‣ NrCubicATGraphs( n ) | ( function ) |
‣ NumberCubicATGraphs( n ) | ( function ) |
Returns: An integer.
Given a positive integer n, this function returns the number of cubic arc-transitive graphs with n vertices stored in this package.
For any positive integers n up to 1280, the current package stores all cubic arc-transitive graphs with n vertices.
gap> NrCubicATGraphs(8192);
93
‣ NrCubicSSGraphs( n ) | ( function ) |
‣ NumberCubicSSGraphs( n ) | ( function ) |
Returns: An integer.
Given a positive integer n, this function returns the number of cubic semisymmetric graphs with n vertices stored in this package.
For any positive integers n up to 1280, the current package stores all cubic semisymmetric graphs with n vertices.
gap> NrCubicSSGraphs(3888);
35
‣ CubicATGraphId( gamma ) | ( attribute ) |
Returns: An integer.
Given a digraph gamma, if gamma is isomorphic to a cubic arc-transitive graph stored in this package, this function returns the index of the graph isomorphic to gamma. Otherwise, this function returns fail.
The index i of a cubic arc-transitive graph gamma in this library is the position at which the graph is stored relative to its number of vertices. In particular, if gamma has n vertices, then gamma will be the ith entry of AllCubicATGraphs(n) and the ith graph found when iterating through CubicATGraphIterator(n).
gap> gamma:=CubicATGraph(8192,40);;
gap> CubicATGraphId(gamma);
40
gap> gamma:=PetersenGraph();;
gap> CubicATGraphId(gamma);
1
‣ CubicSSGraphId( gamma ) | ( attribute ) |
Returns: An integer.
Given a digraph gamma, if gamma is isomorphic to a cubic semisymmetric graph stored in this package, this function returns the index of the graph isomorphic to gamma. Otherwise, this function returns fail.
The index i of a cubic semisymmetric graph gamma in this library is the position at which the graph is stored relative to its number of vertices. In particular, if gamma has n vertices, then gamma will be the ith entry of AllCubicSSGraphs(n) and the ith graph found when iterating through CubicSSGraphIterator(n).
gap> gamma:=CubicSSGraph(6912,20);;
gap> CubicSSGraphId(gamma);
20
‣ SetCubicATGraphProps( gamma ) | ( function ) |
Given a digraph gamma, if this graph is isomorphic to a cubic arc-transitive graph stored in this library, this function sets the properties and attributes of gamma precomputed in this package. This includes
All properties and attributes found in Subsection 3.2-1.
CubicATGraphId (3.2-4).
IsCubicGraph (2.2-6).
IsVertexTransitive (Digraphs: IsVertexTransitive).
gap> gamma:=CompleteDigraph(4);;
gap> SetCubicATGraphProps(gamma);
gap> IsArcTransitiveDigraph(gamma);
true
‣ SetCubicSSGraphProps( gamma ) | ( function ) |
Given a digraph gamma, if this graph is isomorphic to a cubic semisymmetric graph stored in this library, this function sets the properties and attributes of gamma precomputed in this package. This includes
All properties and attributes found in Subsection 3.2-1.
CubicSSGraphId (3.2-5).
IsCubicGraph (2.2-6).
IsVertexTransitive (Digraphs: IsVertexTransitive).
gap> gamma:=CubicSSGraph(6912,25);;
gap> SetCubicSSGraphProps(gamma);
gap> IsVertexTransitive(gamma);
false
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