HaskellForMaths-0.4.7: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Combinatorics.Graph

Description

A module defining a polymorphic data type for (simple, undirected) graphs, together with constructions of some common families of graphs, new from old constructions, and calculation of simple properties of graphs.

Synopsis

# Documentation

set :: Ord b => [b] -> [b] Source

powerset :: [t] -> [[t]] Source

data Graph a Source

Datatype for graphs, represented as a list of vertices and a list of edges. For most purposes, graphs are required to be in normal form. A graph G vs es is in normal form if (i) vs is in ascending order without duplicates, (ii) es is in ascending order without duplicates, (iii) each e in es is a 2-element list [x,y], x<y

Constructors

 G [a] [[a]]

Instances

 Functor Graph Eq a => Eq (Graph a) Ord a => Ord (Graph a) Show a => Show (Graph a)

nf :: Ord a => Graph a -> Graph a Source

Convert a graph to normal form. The input is assumed to be a valid graph apart from order

isSetSystem :: Ord a => [a] -> [[a]] -> Bool Source

isGraph :: Ord a => [a] -> [[a]] -> Bool Source

graph :: Ord t => ([t], [[t]]) -> Graph t Source

Safe constructor for graph from lists of vertices and edges. graph (vs,es) checks that vs and es are valid before returning the graph.

toGraph :: Ord a => ([a], [[a]]) -> Graph a Source

vertices :: Graph t -> [t] Source

edges :: Graph t -> [[t]] Source

incidenceMatrix :: (Num t, Eq a) => Graph a -> [[t]] Source

fromIncidenceMatrix :: (Ord t, Num t, Num a, Eq a, Enum t) => [[a]] -> Graph t Source

adjacencyMatrix :: (Ord a, Num t) => Graph a -> [[t]] Source

fromAdjacencyMatrix :: (Num b, Eq b) => [[b]] -> Graph Int Source

nullGraph :: Integral t => t -> Graph t Source

The null graph on n vertices is the graph with no edges

The null graph, with no vertices or edges

c :: Integral t => t -> Graph t Source

c n is the cyclic graph on n vertices

k :: Integral t => t -> Graph t Source

k n is the complete graph on n vertices

kb :: Integral t => t -> t -> Graph t Source

kb m n is the complete bipartite graph on m and n vertices

kb' :: Integral t => t -> t -> Graph (Either t t) Source

kb' m n is the complete bipartite graph on m left and n right vertices

q :: Integral t => Int -> Graph t Source

q k is the graph of the k-cube

q' :: Integral t => Int -> Graph [t] Source

to1n :: (Ord t, Ord a, Num t, Enum t) => Graph a -> Graph t Source

fromDigits :: Integral a => Graph [a] -> Graph a Source

Given a graph with vertices which are lists of small integers, eg [1,2,3], return a graph with vertices which are the numbers obtained by interpreting these as digits, eg 123. The caller is responsible for ensuring that this makes sense (eg that the small integers are all < 10)

fromBinary :: Integral a => Graph [a] -> Graph a Source

Given a graph with vertices which are lists of 0s and 1s, return a graph with vertices which are the numbers obtained by interpreting these as binary digits. For example, [1,1,0] -> 6.

restriction :: Eq a => Graph a -> [a] -> Graph a Source

The restriction of a graph to a subset of the vertices

inducedSubgraph :: Eq a => Graph a -> [a] -> Graph a Source

lineGraph :: (Ord a, Ord t, Num t, Enum t) => Graph a -> Graph t Source

lineGraph' :: Ord a => Graph a -> Graph [a] Source

cartProd :: (Ord t1, Ord t) => Graph t -> Graph t1 -> Graph (t, t1) Source

valency :: Eq a => Graph a -> a -> Int Source

valencies :: Eq a => Graph a -> [(Int, Int)] Source

valencyPartition :: Eq b => Graph b -> [[b]] Source

isRegular :: Eq t => Graph t -> Bool Source

A graph is regular if all vertices have the same valency (degree)

isCubic :: Eq t => Graph t -> Bool Source

A 3-regular graph is called a cubic graph

nbrs :: Eq a => Graph a -> a -> [a] Source

findPaths :: Eq a => Graph a -> a -> a -> [[a]] Source

distance :: Eq a => Graph a -> a -> a -> Int Source

Within a graph G, the distance d(u,v) between vertices u, v is length of the shortest path from u to v

diameter :: Ord t => Graph t -> Int Source

The diameter of a graph is maximum distance between two distinct vertices

findCycles :: Eq a => Graph a -> a -> [[a]] Source

girth :: Eq t => Graph t -> Int Source

The girth of a graph is the size of the smallest cycle that it contains. Note: If the graph contains no cycles, we return -1, representing infinity.

distancePartition :: Ord a => Graph a -> a -> [[a]] Source

distancePartitionS :: Ord a => [a] -> Set [a] -> a -> [[a]] Source

component :: Ord a => Graph a -> a -> [a] Source

isConnected :: Ord t => Graph t -> Bool Source

Is the graph connected?

components :: Ord a => Graph a -> [[a]] Source

j :: Int -> Int -> Int -> Graph [Int] Source

kneser :: Int -> Int -> Graph [Int] Source

kneser n k returns the kneser graph KG n,k - whose vertices are the k-element subsets of [1..n], with edges between disjoint subsets

gp :: Integral a => a -> a -> Graph (Either a a) Source

prism' :: Integral a => a -> Graph (Either a a) Source