Can we compute the graph Laplacian matrix for a directed graph?

How to find graph laplacian matrix of a directed graph?

Answer 1

[Christine Tobler]

It depends how you want to define it, there is no one consistent definition of what the graph laplacian of a directed graph is. This wikipedia page https://en.wikipedia.org/wiki/Laplacian_matrix#Definitions_for_simple_graphs lists some options.

[Christine Tobler]

For undirected graphs, the graph class in MATLAB has a laplacian method.

[Souarv De]

Thanks @Christine Tobler for providing the resource link for the subject matter. It's really help me a lot.

Answer 2

[Bruno Luong]

In this thread I give formula for graph, for digraph you just need to be careful about indegree or outdegree

% TMW example
s = [1 2 2 3 3 3 4 5 5 5 8 8 9];
t = [2 3 4 1 4 5 5 3 6 7 9 10 10];
G = digraph(s,t);
A = G.adjacency;
% Use Laplacian
Din = diag(sum(A,1)); % in degree matrix
Dout = diag(sum(A,2)); % in degree matrix
Lin = Din - A % laplacian matrix

Lin = 
   (1,1)        1
   (3,1)       -1
   (1,2)       -1
   (2,2)        1
   (2,3)       -1
   (3,3)        2
   (5,3)       -1
   (2,4)       -1
   (3,4)       -1
   (4,4)        2
   (3,5)       -1
   (4,5)       -1
   (5,5)        2
   (5,6)       -1
   (6,6)        1
   (5,7)       -1
   (7,7)        1
   (8,9)       -1
   (9,9)        1
   (8,10)      -1
   (9,10)      -1
  (10,10)       2

Lout = Dout - A % laplacian matrix

Lout = 
   (1,1)        1
   (3,1)       -1
   (1,2)       -1
   (2,2)        2
   (2,3)       -1
   (3,3)        3
   (5,3)       -1
   (2,4)       -1
   (3,4)       -1
   (4,4)        1
   (3,5)       -1
   (4,5)       -1
   (5,5)        3
   (5,6)       -1
   (5,7)       -1
   (8,8)        2
   (8,9)       -1
   (9,9)        1
   (8,10)      -1
   (9,10)      -1

[Souarv De]

Thanks @Bruno Luong for your easy explanation via the simple codding. It's really solved my doubts.