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 \(M^{p \times q}(Z)\).

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 \(P = P_1 \cup P_2 \cup \ldots \cup P_r\) 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 \(P_i\).