HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Combinatorics.GraphAuts

Synopsis

# Documentation

isVertexTransitive :: Ord t => Graph t -> Bool Source #

A graph is vertex-transitive if its automorphism group acts transitively on the vertices. Thus, given any two distinct vertices, there is an automorphism mapping one to the other.

isEdgeTransitive :: Ord t => Graph t -> Bool Source #

A graph is edge-transitive if its automorphism group acts transitively on the edges. Thus, given any two distinct edges, there is an automorphism mapping one to the other.

isArcTransitive :: Ord t => Graph t -> Bool Source #

A graph is arc-transitive (or flag-transitive) if its automorphism group acts transitively on arcs. (An arc is an ordered pair of adjacent vertices.)

isnArcTransitive :: Ord t => Int -> Graph t -> Bool Source #

A graph is n-arc-transitive if its automorphism group is transitive on n-arcs. (An n-arc is an ordered sequence (v0,...,vn) of adjacent vertices, with crossings allowed but not doubling back.)

isDistanceTransitive :: Ord t => Graph t -> Bool Source #

A graph is distance transitive if given any two ordered pairs of vertices (u,u') and (v,v') with d(u,u') == d(v,v'), there is an automorphism of the graph that takes (u,u') to (v,v')

graphAuts :: Ord a => Graph a -> [Permutation a] Source #

Given a graph g, graphAuts g returns a strong generating set for the automorphism group of g.

incidenceAuts :: (Ord p, Ord b) => Graph (Either p b) -> [Permutation p] Source #

Given the incidence graph of an incidence structure between points and blocks (for example, a set system), incidenceAuts g returns a strong generating set for the automorphism group of the incidence structure. The generators are represented as permutations of the points. The incidence graph should be represented with the points on the left and the blocks on the right.

graphAuts7 :: Ord a => Graph a -> ([a], [Permutation a]) Source #

graphAuts8 :: Ord a => Graph a -> ([a], [Permutation a]) Source #

incidenceAuts2 :: (Ord a, Ord b) => Graph (Either a b) -> ([Either a b], [Permutation (Either a b)]) Source #

isGraphAut :: Ord t => Graph t -> Permutation t -> Bool Source #

Is the permutation an automorphism of the graph?

isIncidenceAut :: (Ord p, Ord b) => Graph (Either p b) -> Permutation (Either p b) -> Bool Source #

Is the permutation an automorphism of the incidence structure represented by the graph? (Note that an incidence graph colours points as Left, blocks as Right, and a permutation that swaps points and blocks, even if it is an automorphism of the graph, does not represent an automorphism of the incidence structure. Instead, a point-block crossover is called a duality.)

graphIsos :: (Ord a2, Ord a1) => Graph a1 -> Graph a2 -> [[(a1, a2)]] Source #

incidenceIsos :: (Ord b2, Ord b3, Ord a, Ord b1) => Graph (Either a b1) -> Graph (Either b2 b3) -> [[(a, b2)]] Source #

isGraphIso :: (Ord a, Ord b) => Graph a -> Graph b -> Bool Source #

Are the two graphs isomorphic?

isIncidenceIso :: (Ord p1, Ord b1, Ord p2, Ord b2) => Graph (Either p1 b1) -> Graph (Either p2 b2) -> Bool Source #

Are the two incidence structures represented by these incidence graphs isomorphic?