Talk:Laplacian matrix

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The first definition listed for the Laplacian Matrix, L = D-A seems to not match the second definition. With the example graph below, D(1,1) = 4 and A(1,1) = 1 (since there is a loop connecting vertex 1 with itself). Then, if L = D - A, we would have L(1,1) = 4 - 1 = 3. But in fact, L(1,1) =4.

I checked, and it only mentions the second definition. Therefore, I'm removing the definition L = D - A. (Georgevulov 23:01, 18 August 2007 (UTC))

From the literature it looks like only a few electrical engineering type people call this an Admittance Matrix, and everybody else calls it a Laplacian Matrix. Does anyone else have an opinion on this?

Meekohi 01:19, 15 December 2005 (UTC)

I have never seen this called anything but the Laplacian matrix in the mathematics literature. JLeander 18:53, 26 August 2006 (UTC)

Spectral gap[edit]

The article currently states:

The smallest non-trivial eigenvalue of L is called the spectral gap or Fiedler value.

Yet the article on expander graphs states that the spectral gap is the difference between the two largest eigenvalues of the adjacency matrix. That these two might be the same thing is not obvious, and needs clarification. linas (talk) 23:49, 7 September 2008 (UTC)


The link to fdm should be disambiguated. I don't know where it should go though. (talk) 08:36, 20 January 2012 (UTC)


The etymology needs to be included. (talk) 06:56, 10 November 2014 (UTC)

Inconsistent with degree matrix[edit]

In the degree matrix definition a directed graph is given as the example. In this the example is the undirected graph.

I think a note to flag this is important. (talk) 13:29, 11 August 2015 (UTC)

The definition here of incidence matrix appears to be transpose of the one from the page devoted to incidence matrix86.163.130.180 (talk) 22:11, 20 May 2016 (UTC)

The name of the random-walk normalized Laplacian[edit]

I quote: "The name of the random-walk normalized Laplacian comes from the fact that this matrix is simply the transition matrix of a random walker on the graph.". This can't be true, as a transition matrix is nonnegative. --Peleg (talk) 12:50, 27 October 2015 (UTC)

Definition of random-walk normalized Laplacian[edit]

The definition of random walk normalized Laplacian is inconsistent with the one given in the beginning. It is also different from the element-wise definition that follows (as is non negative). It seems that it should be defined as , however, the identity involving , is correct according to the current definition. — Preceding unsigned comment added by (talk) 16:49, 9 November 2015 (UTC)

I think the element-wise definition is wrong, since for the random walk the probability of going to from node to any other adjacent node is uniform, so and definitely not negative! — Preceding unsigned comment added by (talk) 08:49, 9 September 2016 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Laplacian matrix/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

In the "Definition" section ui and uj are not defined

Last edited at 09:59, 13 May 2009 (UTC). Substituted at 02:16, 5 May 2016 (UTC)

Unreliable reference[edit]

People, MathWorld is not a reliable reference. It has enough errors that you cannot simply trust it. (I rarely look at MathWorld but when I do, I find errors fairly often.) I propose it be removed from this article.

For the definition of the Laplacian matrix, it is in so many textbooks that it could be treated as common knowledge. Source less elementary statements, I would say (IMO, but I may be wrong about this). Zaslav (talk) 22:11, 23 August 2016 (UTC)