In this chapter we give functions for accessing the locally arc-transitive \(\{{3,4}\}\)-valent graphs stored in this package, and related properties.
In this package, we collect bipartite biregular graphs of valence \(\{{3,4}\}\) that are locally \(G\)-arc-transitive for some group of automorphisms \(G\) with the edge stabiliser acting faithfully on the union of the neighbourhoods of the vertices of the edge. Currently, this package stores all such graphs on up to 350 vertices, and many more on up to 1050. For more information and references about these graphs, see [Pot12].
In this Section we introduce functions for the access to the locally arc-transitive \(\{{3,4}\}\)-valent graphs stored in the GraphSym package.
‣ LAT34ValentGraph( 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 SetLAT34ValentGraphProps (10.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graph.
gap> LAT34ValentGraph(896,200);
<immutable symmetric digraph with 896 vertices, 3072 edges>
gap> LAT34ValentGraph(896,300);
fail
‣ AllLAT34ValentGraphs( 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 SetLAT34ValentGraphProps (10.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> gammas:=AllLAT34ValentGraphs(224);;
gap> Length(gammas);
22
‣ LAT34ValentGraphIterator( 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 SetLAT34ValentGraphProps (10.2-3). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> cnt:=0;; iter:=LAT34ValentGraphIterator(224);;
gap> for gamma in iter do
> if HasSolvableAutGroup(gamma) then
> cnt:=cnt+1;
> fi;
> od;
gap> cnt;
22
In this Section we give the functions which give information about the locally arc-transitive \(\{{3,4}\}\)-valent graph library, and the properties and attributes of the graphs it contains.
‣ NrLAT34ValentGraphs( n ) | ( function ) |
‣ NumberLAT34ValentGraphs( 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> NrLAT34ValentGraphs(896);
249
‣ LAT34ValentGraphId( 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 AllLAT34ValentGraphs(n) and the ith graph found when iterating through LAT34ValentGraphIterator(n).
gap> gamma:=LAT34ValentGraph(896,20);;
gap> LAT34ValentGraphId(gamma);
20
‣ SetLAT34ValentGraphProps( 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:
LAT34ValentGraphId (10.2-2).
gap> gamma:=LAT34ValentGraph(896,20);;
gap> SetLAT34ValentGraphProps(gamma);
gap> LAT34ValentGraphId(gamma);
20
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