# How to find the laplacian matrix of a weighted directed graph

How to find the laplacian matrix of a weighted directed graph? Please explain with an example

How to find the laplacian matrix of a weighted directed graph

**
asked 2017-10-31 08:16:30 +0200 **

Anonymous

How to find the laplacian matrix of a weighted directed graph? Please explain with an example

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1

```
sage: G = DiGraph(3, weighted=True)
sage: G.add_edges([(0,2,4), (1,2,6),(0,1,2)])
sage: G.laplacian_matrix()
[ 0 -2 -4]
[ 0 2 -6]
[ 0 0 10]
```

0

This is an alternative longer answer...

One problem may be, depending on the situation, to initialize and/or declare the "weight data" for the di(rected )graph.
The following assumes that the weights are given in a matrix `A`

, then it constructs in one line the corresponding digraph. The method `laplacian_matrix`

, applied on it gives the result. Sample code follows. Instead of typing myself the entries of a matrix, there will be a random generation.

```
import random
random.seed( 20171101 )
def myrandomizer():
if random.uniform( 0,1 ) < 0.7:
return 0
return random.choice( [ 1..10 ] )
N = 8
A = matrix( QQ, N, N, [ myrandomizer() for _ in range(N^2) ] )
print A
D = DiGraph( A, format='weighted_adjacency_matrix' )
# D.show() # decomment to show...
```

The only important line above, given `A`

is the one with `D = DiGraph( A, format='weighted_adjacency_matrix' )`

. (Without specifying the format, `sage`

is trying to guess the format, and it rather generates multiple edges between two vertices for a (weight) entry $\ge 2$.)

The `print A`

line gives the reproducible random matrix

```
[ 7 0 0 0 0 3 2 0]
[ 0 0 0 0 7 10 0 0]
[ 0 0 0 0 0 6 3 0]
[10 0 6 0 0 0 0 0]
[ 8 7 9 3 0 3 0 9]
[ 0 0 5 0 0 0 0 0]
[ 0 5 0 0 0 0 0 2]
[ 0 0 0 0 3 6 0 0]
sage:
```

In the given situation we can apply the following related methods:

```
sage: D.laplacian_matrix()
[ 18 0 0 0 0 -3 -2 0]
[ 0 12 0 0 -7 -10 0 0]
[ 0 0 20 0 0 -6 -3 0]
[-10 0 -6 3 0 0 0 0]
[ -8 -7 -9 -3 10 -3 0 -9]
[ 0 0 -5 0 0 28 0 0]
[ 0 -5 0 0 0 0 5 -2]
[ 0 0 0 0 -3 -6 0 11]
sage: D.adjacency_matrix()
[1 0 0 0 0 1 1 0]
[0 0 0 0 1 1 0 0]
[0 0 0 0 0 1 1 0]
[1 0 1 0 0 0 0 0]
[1 1 1 1 0 1 0 1]
[0 0 1 0 0 0 0 0]
[0 1 0 0 0 0 0 1]
[0 0 0 0 1 1 0 0]
sage: D.weighted()
True
sage: D.weighted_adjacency_matrix()
[ 7 0 0 0 0 3 2 0]
[ 0 0 0 0 7 10 0 0]
[ 0 0 0 0 0 6 3 0]
[10 0 6 0 0 0 0 0]
[ 8 7 9 3 0 3 0 9]
[ 0 0 5 0 0 0 0 0]
[ 0 5 0 0 0 0 0 2]
[ 0 0 0 0 3 6 0 0]
sage:
```

Note that the laplacian matrix differs from minus the weighted adjacency matrix only on the diagonal. The diagonal entries are adjusted, so that all sums of the **columns** vanish. Some books define the laplacian so that the sums on the rows vanish.

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Asked: ** 2017-10-31 08:16:30 +0200 **

Seen: **638 times**

Last updated: **Nov 01 '17**

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