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

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Souarv De on 16 Mar 2022
Commented: Souarv De on 16 Mar 2022
How to find graph laplacian matrix of a directed graph?

Christine Tobler on 16 Mar 2022
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 on 16 Mar 2022
For undirected graphs, the graph class in MATLAB has a laplacian method.
Souarv De on 16 Mar 2022
Thanks @Christine Tobler for providing the resource link for the subject matter. It's really help me a lot.

Bruno Luong on 16 Mar 2022
Edited: Bruno Luong on 16 Mar 2022
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);
% 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 on 16 Mar 2022
Thanks @Bruno Luong for your easy explanation via the simple codding. It's really solved my doubts.

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