Matrices and Vector Spaces Associated with a Graph or Digraph

AdjacencyMatrix(G) : Grph -> AlgMatElt

Returns the adjacency matrix for the (p, q) graph G as an element of the matrix ring M_p(Z).

DistanceMatrix(G) : Grph -> AlgMatElt

Returns the distance matrix A for the (p, q) graph G as an element of the matrix ring M_p(Z). The (i, j)-th entry of A gives the distance between vertices v_i and v_j of G.

IncidenceMatrix(G) : Grph -> ModHomElt

Returns the incidence matrix M for the (p, q) graph G as an element of the matrix bimodule M^{p x q}(Z).

If G is a graph, then entry (i, j) of M is 1 if the vertex v_i of G lies on the edge e_j of G. Otherwise entry (i, j) is zero.

If G is a digraph, entry (i, j) of M is 1 if vertex v_i is the initial vertex of the edge e_j, and -1 if v_i is the final vertex of the edge e_j. Otherwise entry (i, j) is zero. If e_j is a loop, then entry (i, j) may be either 1 or -1.

IntersectionMatrix(G, P) : GrphUnd, { { GrphVert } } -> AlgMatElt

IntersectionMatrix(G, P) : GrphUnd, { { RngIntElt } } -> AlgMatElt

Given an ordered equitable partition P = P₁ ∪P₂ ∪ ... ∪P_r of the vertex-set of the graph G, return the intersection matrix T for the partition. Thus, entry T[i, j] is the number of vertices of the set P_j that are adjacent to a vertex of the set P_i.