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WeightedAdjacencyMatrix

WeightedAdjacencyMatrix

WeightedAdjacencyMatrix[g]

gives the adjacency matrix of edge weights of the graph g.

WeightedAdjacencyMatrix[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

WeightedAdjacencyMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.
An entry w_ij of the weighted adjacency matrix is the weight of a directed edge from vertex ν_i to vertex ν_j. If there is no edge the weight is taken to be 0.

An edge without explicit EdgeWeight specified is taken to have weight 1.
An undirected edge is interpreted as two directed edges with opposite directions and the same weight.
The vertices v_i are assumed to be in the order given by VertexList[g].
The weighted adjacency matrix for a graph will have dimensions ×, where is the number of vertices.

Examples

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Basic Examples (2)

The weighted adjacency matrix of an undirected graph:

Wolfram Language code: Graph[{12, 23, 31}, EdgeWeight -> RandomReal[{1, 5}, 3]]

Wolfram Language code: WeightedAdjacencyMatrix[%]//MatrixForm

The weighted adjacency matrix of a directed graph:

Wolfram Language code: Graph[{12, 23, 31}, EdgeWeight -> RandomReal[{1, 5}, 3]]

Wolfram Language code: WeightedAdjacencyMatrix[%]//MatrixForm

Scope (5)

The weighted adjacency matrix of an undirected graph is symmetric:

Wolfram Language code: Graph[{12, 13, 23, 24, 34}, EdgeWeight -> RandomReal[{1, 5}, 5]]

Wolfram Language code: WeightedAdjacencyMatrix[%]//MatrixForm

The weighted adjacency matrix of a directed graph can be unsymmetric:

Wolfram Language code: Graph[{12, 21, 31, 32, 41, 42}, EdgeWeight -> RandomReal[{1, 5}, 6]]

Wolfram Language code: WeightedAdjacencyMatrix[%]//MatrixForm

Use rules to specify the graph:

Wolfram Language code: WeightedAdjacencyMatrix[{1 -> 3, 2 -> 1}]//MatrixForm

The weighted adjacency matrix of the graph with self-loops has diagonal entries:

Wolfram Language code: Graph[{11, 12, 23, 31}, EdgeWeight -> RandomReal[{1, 5}, 4]]

Wolfram Language code: WeightedAdjacencyMatrix[%]//MatrixForm

WeightedAdjacencyMatrix works with large graphs:

Wolfram Language code: Graph[Table[i -> Mod[i ^ 2, 10 ^ 3], {i, 0, 10 ^ 3 - 1}]];

Wolfram Language code: Timing[m = WeightedAdjacencyMatrix[%]]

Use MatrixPlot to visualize the matrix:

Wolfram Language code: MatrixPlot[m]

Properties & Relations (4)

Rows and columns of the weighted adjacency matrix follow the order given by VertexList:

Wolfram Language code: g = Graph[{23, 31, 12, 14}, EdgeWeight -> RandomReal[{1, 5}, 4], VertexShapeFunction -> "Name"]

Wolfram Language code: VertexList[g]

Wolfram Language code: TableForm[Normal@WeightedAdjacencyMatrix[g], TableHeadings -> {c = Style[#, Blue]& /@ %, c}]

Use WeightedAdjacencyGraph to construct a graph from a weighted adjacency matrix:

Wolfram Language code: WeightedAdjacencyGraph[{{∞, 1, ∞}, {∞, ∞, 1}, {1, 1, ∞}}]

Wolfram Language code: WeightedAdjacencyMatrix[%]//Normal

The number of rows or columns is equal to the number of vertices:

Wolfram Language code: g = CompleteGraph[5, EdgeWeight -> RandomReal[{1, 5}, 10]]

Wolfram Language code: Dimensions[AdjacencyMatrix[g]]

Wolfram Language code: VertexCount[g]

The main diagonals for a loop-free graph are all zeros:

Wolfram Language code: GridGraph[{2, 3}, EdgeWeight -> RandomReal[{1, 5}, 7]]

Wolfram Language code: Diagonal[AdjacencyMatrix[%]]//Normal

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