In this Chapter we give functions for accessing the half-arc-transitive 4-valent graphs stored in this package, and related properties. Currently, this package contains all half-arc-transitive 4-valent graphs on up to 1000 vertices. For information and references on these graphs, see [PSV13a].
Let \(\Gamma\) be a simple graph (undirected, loopless, without multiple edges). Then \(\Gamma\) is 4-valent (or tetravalent) if each vertex of the graph has exactly 4 neighbours.
The graph \(\Gamma\) is half-arc-transitive if the automorphism group of \(\Gamma\) acts transitively on the vertices and edges of \(\Gamma\), but not on the arcs of \(\Gamma\).
In this Section we introduce functions for the access to the half-arc-transitive 4-valent graphs stored in the GraphSym package.
‣ HAT4ValentGraph( n, i[, data] ) | ( function ) |
Returns: A digraph.
Given positive integers n,i, this function returns the ith half-arc-transitive 4-valent 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 SetHAT4ValentGraphProps (6.2-4). If data=false or not specified, none of these properties or attributes are given to the resulting graph.
gap> HAT4ValentGraph(600,20);
<immutable symmetric digraph with 600 vertices, 2400 edges>
gap> HAT4ValentGraph(600,28);
fail
‣ AllHAT4ValentGraphs( n[, data] ) | ( function ) |
Returns: A list.
Given a positive integer n, this function returns a list containing all half-arc-transitive 4-valent 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 SetHAT4ValentGraphProps (6.2-4). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> gammas:=AllHAT4ValentGraphs(600);;
gap> Length(gammas);
27
‣ HAT4ValentGraphIterator( n[, data] ) | ( function ) |
Returns: An iterator.
Given a positive integer n, this function returns an iterator over all half-arc-transitive 4-valent 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 SetHAT4ValentGraphProps (6.2-4). If data=false or not specified, none of these properties or attributes are given to the resulting graphs.
gap> cnt:=0;; iter:=HAT4ValentGraphIterator(600);;
gap> for gamma in iter do
> if HasSolvableAutGroup(gamma) then
> cnt:=cnt+1;
> fi;
> od;
gap> cnt;
25
In this Section we give the functions which give information about the half-arc-transitive 4-valent graph library, and the properties and attributes of the graphs it contains.
For a given half-arc-transitive 4-valent graph \(\Gamma\) stored in this package, there are several precomputed attributes stored in the GraphSym package. These include the following:
Girth of \(\Gamma\) (DigraphUndirectedGirth (Digraphs: DigraphUndirectedGirth)).
Bipartiteness of \(\Gamma\) (IsBipartiteDigraph (Digraphs: IsBipartiteDigraph)).
Arc-transitivity of \(\Gamma\) (IsArcTransitiveDigraph (2.2-8)).
The Cayleyness of \(\Gamma\) (IsCayleyGraph (2.2-7)).
The length of the shortest consistent cycle in \(\Gamma\) (see ConsistentCycleTypes (5.2-6)).
The length of the longest consistent cycle in \(\Gamma\) (see ConsistentCycleTypes (5.2-6)).
Now we introduce functions which are used to find information about the library and each of the graphs it stores.
‣ NrHAT4ValentGraphs( n ) | ( function ) |
‣ NumberHAT4ValentGraphs( n ) | ( function ) |
Returns: An integer.
Given a positive integer n, this function returns the number of half-arc-transitive 4-valent graphs with n vertices stored in this package.
For any positive integers \(n\) up to 1000, the current package stores all half-arc-transitive 4-valent graphs with \(n\) vertices.
gap> NrHAT4ValentGraphs(600);
27
‣ HAT4ValentGraphId( 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 AllHAT4ValentGraphs(n) and the ith graph found when iterating through HAT4ValentGraphIterator(n).
gap> gamma:=HAT4ValentGraph(768,20);;
gap> HAT4ValentGraphId(gamma);
20
‣ SetHAT4ValentGraphProps( 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
All properties and attributes found in Subsection 6.2-1.
HAT4ValentGraphId (6.2-3).
IsCayleyGraph (2.2-7).
IsVertexTransitive (Digraphs: IsVertexTransitive).
gap> gamma:=HAT4ValentGraph(768,20);;
gap> SetHAT4ValentGraphProps(gamma);
gap> SizeStabAut(gamma);
8
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