In this chapter we give functions for accessing the edge-transitive 4-valent graphs stored in this package, and related properties. Currently, this package contains an incomplete list of edge-transitive 4-valent graphs on up to 512 vertices. For information and references about these graphs, see [PW16].
Let Γ be a simple graph (undirected, loopless, without multiple edges). Then Γ is 4-valent (or tetravalent) if each vertex of the graph has exactly 4 neighbours.
The graph Γ is edge-transitive if the automorphism group of Γ acts transitively on the edges of Γ.
In this Section we introduce functions for the access to the edge-transitive 4-valent graphs stored in the GraphSym package.
‣ ET4ValentGraph( n, i[, data] ) | ( function ) |
Returns: A digraph.
Given positive integers n,i, this function returns the ith arc-transitive 2-valent digraph 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 SetET4ValentGraphProps (9.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graph.
gap> ET4ValentGraph(512,100);
<immutable symmetric digraph with 512 vertices, 2048 edges>
gap> ET4ValentGraph(512,700);
fail
‣ AllET4ValentGraphs( n[, data] ) | ( function ) |
Returns: A list.
Given a positive integer n, this function returns a list containing all arc-transitive digraphs 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 SetET4ValentGraphProps (9.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> gammas:=AllET4ValentGraphs(400);;
gap> Length(gammas);
107
‣ ET4ValentGraphIterator( n[, data] ) | ( function ) |
Returns: An iterator.
Given a positive integer n, this function returns an iterator over all arc-transitive 2-valent digraphs 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 SetET4ValentGraphProps (9.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> cnt:=0;; iter:=ET4ValentGraphIterator(400);;
gap> for gamma in iter do
> if HasSolvableAutGroup(gamma) then
> cnt:=cnt+1;
> fi;
> od;
gap> cnt;
106
In this Section we give the functions which give information about the edge-transitive 4-valent graph library, and the properties and attributes of the graphs it contains.
‣ NrET4ValentGraphs( n ) | ( function ) |
‣ NumberET4ValentGraphs( n ) | ( function ) |
Returns: An integer.
Given a positive integer n, this function returns the number of arc-transitive 2-valent digraphs with n vertices stored in this package.
For any positive integers n up to 1000, the current package stores all arc-transitive 2-valent digraphs with n vertices.
gap> NrET4ValentGraphs(400);
107
‣ ET4ValentGraphId( gamma ) | ( attribute ) |
Returns: An integer.
Given a digraph gamma, if gamma is isomorphic to a 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 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 AllET4ValentGraphs(n) and the ith graph found when iterating through ET4ValentGraphIterator(n).
gap> gamma:=CompleteDigraph(5);;
gap> ET4ValentGraphId(gamma);
1
gap> gamma:=ET4ValentGraph(512,40);;
gap> ET4ValentGraphId(gamma);
40
‣ SetET4ValentGraphProps( gamma ) | ( function ) |
Given a digraph gamma, if this graph is isomorphic to a graph stored in this library, this function sets the properties and attributes of gamma precomputed in this package. This includes:
ET4ValentGraphId (9.2-2).
gap> gamma:=ET4ValentGraph(512,40);;
gap> SetET4ValentGraphProps(gamma);
gap> ET4ValentGraphId(gamma);
40
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