datatype=type

Specifies a datatype for the generated Matrix as described in rtable. The default is datatype=anything.

order=one of C_order or Fortran_order

Specifies the order of the generated Matrix. The default is order=C_order.

shape=name or list of names

Specifies the storage allocation for Matrix entries. Must be a name or list of names specifying one or more built-in or user-defined indexing functions. The default is shape=symmetric if G is undirected and shape=[] if G is directed.

ModularityMatrix(G) returns the modularity matrix of a graph G.

The default output is an n by n Matrix with the following properties:

By default datatype=anything and order=C_order

If G is undirected, by default shape=symmetric

If G is directed, by default shape=[]

Let G be a graph and let m be its number of edges. The modularity matrix of G is defined as follows:

If G is undirected, the entry $\left(i\,j\right)$ of this matrix is deg[i]*deg[j]/2/m where deg[i] is the degree of vertex i.

If G is directed, the entry $\left(i\,j\right)$ of this matrix is indeg[i]*outdeg[j]/m where indeg[i] is the in-degree of vertex i and outdeg[j] is the out-degree of vertex j.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(\left[1\,2\,3\,4\right]\,\mathrm{Trail}\left(1\,2\,3\,4\,1\right)\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 4\; edges}$

$\mathrm{ModularityMatrix}\left(G\right)$

$\left[\begin{array}{cccc}-\frac{1}{2}& \frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}& \frac{1}{2}& -\frac{1}{2}\\ -\frac{1}{2}& \frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}& \frac{1}{2}& -\frac{1}{2}\end{array}\right]$

The GraphTheory[ModularityMatrix] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.