Subsections

• Graphs Constructed from Groups
• Graphs Constructed from Codes and Designs

Given a finite group A defined on generating set X, construct the Cayley graph C of A relative to the generating set X. This graph is defined as follows: The vertices correspond to the elements of A and two vertices u, v are adjacent if and only if there exists an element x in X such that u * x = v.

[Future release] CayleyGraph(A, X) : Grp, { GrpElt } -> GrphUnd

Given a finite group A, and a set X of elements of G which is closed under the taking of inverses, construct the Cayley graph C of A relative to the set X. This graph is defined as follows: The vertices correspond to the elements of A and two vertices u, v are adjacent if and only if there exists an element x in X such that u * x = v.

Given a finite group A defined on the generating set X and a subgroup B of A, construct the Schreier coset graph S for A over B, relative to X. The graph S is defined as follows: The vertices correspond to the cosets of B in A, and two vertices u, v are adjacent in S if and only if there exists an element x in X such that u * x = v.

[Future release] SchreierGraph(A, B, X) : Grp, Grp, { GrpElt } -> GrphUnd

Given a finite group A, a subgroup B of A, and a set X of elements of G closed under the taking of inverses, construct the Schreier coset graph S for A over B, relative to X. The graph S is defined as follows: The vertices correspond to the cosets of B in A, and two vertices u, v are adjacent in S if and only if there exists an element x in X such that u * x = v.

Let P be a transitive permutation group acting on the set Omega = {1, ..., n}. Let u be an element of Omega and let T = {t_1, ..., t_r} be a subset of Omega. This function constructs the digraph G corresponding to the union of P-orbits containing the pairs (u, t_1), ..., (u, t_r).

Let P be a transitive permutation group acting on the set Omega = {1, ..., n}. Let u be an element of Omega and let T = {t_1, ..., t_r} be a subset of Omega. This function constructs the underlying graph G of the digraph corresponding to the union of P-orbits containing the pairs (u, t_1), ..., (u, t_r). Thus, if T defines a self-paired orbit Delta of the stabilizer in P of the point u, this function constructs the orbital graph associated with Delta.

Let P be a permutation group acting on the set Omega = {1, ..., n}. Let G be a graph (digraph) with vertices v_1, ..., v_n. This function adds the minimum number of edges to G so as to produce a graph (digraph) H which is left invariant by the group P.

Given a code C and a positive integer d, construct the graph whose vertices are the codewords of C with codewords u and v adjacent if and only the Hamming distance between u and v is d.

Given a linear code C, construct the graph G whose vertices are the cosets of C. Two cosets C_i and C_j are adjacent if and only if there is a code word u in C_i and a codeword v in C_j such that the Hamming distance between u and v is one.

Given an incidence structure D = (X, B), construct the incidence graph G of D. The vertices of G is X union B. The adjacency rules are as follows: No two vertices of X are adjacent; no two vertices of B are adjacent; a vertex x in X is adjacent to a vertex b in B if and only if x is in b.

Given an incidence structure D = (X, B), construct the point graph G of D. The vertex set of G is X. Vertices x in X, y in X are adjacent in G if there is a block b in B such that x in b and y in b.

The block graph of D; i.e. the point graph of the dual of D.

Given a plane P with point-set V and line-set L, construct the incidence graph G of P. The vertex-set of G is V union L. The adjacency rules are as follows: No two vertices of V are adjacent; no two vertices of L are adjacent; a vertex v in V is adjacent to a vertex a in L if and only if v lies on a.

Given a plane P with point-set V and line-set L, construct the point graph G of P. The vertex-set of G is V. Vertices u, v in V are adjacent in G iff there is a line in L that contains them both.

Given a plane P with point-set V and line-set L, construct the point graph G of P. The vertex-set of G is L. Lines a, b in L are adjacent in G iff there is a vertex in V that lies on them both.