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How can I decide which are the connected components by looking at the eigenvectors of laplacian matrix ?

I have the following adjacency matrix (all nodes are different), and have created the Laplacian matrix. I calculated the eigenvalues:

$λ_1 = 0$, $λ_2 = 0$, $λ_3 = 1$ which mean there is 2 connected components.

The eigenvectors are :

(0,x2,x3) and (-x2,x2,0) as calculated using this site:

https://matrixcalc.org/en/vectors.html

I was wondering, how can I find the connected components using the eigenvectors?

David G. Stork
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user5132182
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1 Answers1

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You can find the connected components of the graph from the kernel of the Laplacian matrix, i.e., the eigenspace of $0$. Note that you didn’t need to find all of the eigenvalues and eigenvectors to do this. You could’ve simply row-reduced the matrix (see here for an example and the earlier question linked to it for a general explanation).

In your case, you’ve already found one of the basis vectors, which tells you that one of the connected components consists of the second and third nodes. Since you know that there are only two connected components, the other one must be the complement of this set.

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amd
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