BUILT-IN MATHEMATICA SYMBOL

KirchhoffMatrix

KirchhoffMatrix[g]
gives the Kirchhoff matrix of the graph g.

KirchhoffMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.

The Kirchhoff matrix for a graph will have dimensions × where is the number of vertices.

The Kirchhoff matrix of an undirected graph:

The Kirchhoff matrix of a directed graph:

The Kirchhoff matrix of an undirected graph:

Out[1]=

Out[2]//MatrixForm=

The Kirchhoff matrix of a directed graph:

Out[1]=

Out[2]//MatrixForm=

The Kirchhoff matrix of an undirected graph is symmetric:

The Kirchhoff matrix of a directed graph can be unsymmetric:

The Kirchhoff matrix of a non-simple graph and its simple graph is the same:

KirchhoffMatrix works with large graphs:

Use MatrixPlot to visualize the matrix:

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

Use KirchhoffMatrix to construct a graph from a Kirchhoff matrix:

The degree of vertices can be found using the diagonal of the Kirchhoff matrix:

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

The off-diagonal entries of a Kirchhoff matrix are

or

:

For a complete graph, all entries outside the diagonal are

in the Kirchhoff matrix:

A complete

-partite graph has

off-diagonal block entries:

In particular, TuranGraph and StarGraph are bipartite:

A path graph will have

or

in the diagonal and

off-diagonal bands: