Add (has_)disjoint_automorphisms, petersen_graph, and clebsch_graph

src/Combinatorics/Graphs/functions.jl

@@ -2164,3 +2164,93 @@ end
 
 _is_tree(g::Graph{Directed}) = is_weakly_connected(g) && isone(n_vertices(g) - n_edges(g))
 
+
+@doc raw"""
+    petersen_graph()
+
+Construct and return the Petersen graph as a simple undirected graph.
+
+"""
+function petersen_graph()
+  e = Vector{Int}[[1,3],[1,4],[2,4],[2,5],[3,5],[1,6],[6,7],[2,7],[7,8],[3,8],[8,9],[4,9],[9,10],[5,10],[10,6]]
+  return graph_from_edges(e)
+end
+
+@doc raw"""
+    clebsch_graph()
+
+Construct and return the 5-regular Clebsch graph.
+
+The Clebsch graph is a strongly regular graph with 16 vertices and 40 edges. It is triangle-free, has degree 5.
+"""
+function clebsch_graph()
+  e = Vector{Int}[[1, 2], [1, 3], [1, 5], [1, 9], [1, 16], [2, 4], [2, 6], [2, 10], [2, 15], [3, 4], [3, 7], [3, 11], [3, 14], [4, 8], [4, 12], [4, 13], [5, 6], [5, 7], [5, 13], [5, 12], [6, 8], [6, 14], [6, 11], [7, 8], [7, 15], [7, 10], [8, 16], [8, 9], [9, 10], [9, 11], [9, 13], [10, 12], [10, 14], [11, 12], [11, 15], [12, 16], [13, 14], [13, 15], [14, 16], [15, 16]]
+  return graph_from_edges(e)
+end
+
+@doc raw"""
+    disjoint_automorphisms(G::Graph)
+
+Find and return a pair of automorphisms of the graph `G` which are disjoint and
+neither is the identity (thus neither fixes all vertices; and hence, since they
+are disjoint, neither moves all vertices). Returns a tuple `(g1, g2)` of such
+automorphisms if found, otherwise throws an error.
+
+Two autormorphisms $\sigma$ and $\tau$ are said to be disjoint if
+$\sigma(i) = i$ if and only if $\tau(i) \neq i$ for all vertices $i$ of the graph.
+
+# Examples
+```jldoctest
+julia> C = clebsch_graph();
+
+julia> disjoint_automorphisms(C)
+((2,3)(6,7)(10,11)(14,15), (1,4)(5,8)(9,12)(13,16))
+```
+"""
+function disjoint_automorphisms(G::Graph)
+  A = automorphism_group(G)
+  n = nv(G)
+
+  ret, a, b = _compute_disjoint_automorphism(G::Graph)
+  @req ret "The graph has no disjoint automorphisms"
+  return a, b
+end
+
+@doc raw"""
+    has_disjoint_automorphisms(G::Graph)
+
+Return `true` if the graph `G` has a pair of non-trivial, disjoint automorphisms,
+and `false` otherwise.
+
+# Examples
+```jldoctest
+julia> C = petersen_graph();
+
+julia> has_disjoint_automorphisms(C)
+false
+```
+"""
+function has_disjoint_automorphisms(G::Graph)
+  A = automorphism_group(G)
+  n = nv(G)
+  ret, a, b = _compute_disjoint_automorphism(G::Graph)
+  return ret
+end
+
+@attr Tuple{Bool, PermGroupElem, PermGroupElem} function _compute_disjoint_automorphism(G::Graph)
+  A = automorphism_group(G)
+  n = nv(G)
+
+  for cc in conjugacy_classes(A)
+    rcc = first(cc)
+    rcc == one(A) && continue
+    stab =  stabilizer(A, moved_points(rcc))[1]
+    order(stab) > 1 || continue
+    for g2 in stab
+      g2 == one(A) && continue
+      return (true, rcc, g2)
+    end
+  end
+  return (false, one(A), one(A))
+end
+

src/exports.jl

@@ -409,6 +409,7 @@ export class_positions_of_pcore
 export class_positions_of_solvable_residuum
 export closed_subvariety_of_toric_variety
 export closure
+export clebsch_graph
 export cm_regularity
 export coatoms
 export cobases
@@ -559,6 +560,7 @@ export direct_sum
 export direct_sum_components
 export directed_component
 export discriminant_representation
+export disjoint_automorphisms, has_disjoint_automorphisms
 export disjoint_union
 export div_left
 export div_left!
@@ -1433,6 +1435,7 @@ export permutation_group
 export permutation_matrix
 export permutation_of_terms
 export permuted
+export petersen_graph
 export phylogenetic_tree
 export picard_class
 export picard_group

test/Combinatorics/Graph.jl

@@ -292,4 +292,33 @@
       @test sg4.vertexlabels[1] == "second"
       @test sg4.vertexlabels[2] == "fourth"
     end
+
+    @testset "petersen_graph" begin
+      P = petersen_graph()
+      @test n_vertices(P) == 10
+      @test n_edges(P) == 15
+      @test degree(P) == fill(3,10)
+    end
+
+    @testset "clebsch_graph" begin
+      C = clebsch_graph()
+      @test n_vertices(C) == 16
+      @test n_edges(C) == 40
+      @test degree(C) == fill(5,16)
+    end
+
+    @testset "disjoint automorphism" begin
+      P = petersen_graph()
+      @test !has_disjoint_automorphisms(P)
+      @test_throws ArgumentError disjoint_automorphisms(P)
+
+      C = clebsch_graph()
+      @test has_disjoint_automorphisms(C)
+      a,b = disjoint_automorphisms(C)
+      G = automorphism_group(C)
+      @test a != one(G)
+      @test b != one(G)
+      @test fixed_points(a) == moved_points(b)
+      @test fixed_points(b) == moved_points(a)
+    end
 end