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

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How to find graph laplacian matrix of a directed graph?

Accepted Answer

Christine Tobler
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.
  2 Comments
Souarv De
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.

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More Answers (1)

Bruno Luong
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);
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

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