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

Question

Souarv De 2022-3-16

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https://ww2.mathworks.cn/matlabcentral/answers/1672949-can-we-compute-the-graph-laplacian-matrix-for-a-directed-graph

评论： Souarv De 2022-3-16

采纳的回答： Christine Tobler

How to find graph laplacian matrix of a directed graph?

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Answer 1

Christine Tobler 2022-3-16

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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.

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Christine Tobler 2022-3-16

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

Souarv De 2022-3-16

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

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Answer 2

Bruno Luong 2022-3-16

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https://ww2.mathworks.cn/matlabcentral/answers/1672949-can-we-compute-the-graph-laplacian-matrix-for-a-directed-graph#answer_919139

编辑：Bruno Luong 2022-3-16

在 MATLAB Online 中打开

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