Talk:Laplacian matrix

Symmetric Laplacian via the incidence matrix

The text ", with i < j" is redundant and can be omitted. $L=BB^{\textsf {T}}$ is independent of the direction of the edges. KP (talk) 17:18, 7 March 2024 (UTC)

Untitled

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 wolfram.com, 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)

Fixed in the current revision 24.61.244.65 (talk) 01:56, 1 February 2022 (UTC)

Spectral gap

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)

Fixed in the current revision 24.61.244.65 (talk) 02:13, 1 February 2022 (UTC)

fdm

The link to fdm should be disambiguated. I don't know where it should go though. 131.246.194.40 (talk) 08:36, 20 January 2012 (UTC)

Fixed in the current revision 24.61.244.65 (talk) 01:57, 1 February 2022 (UTC)

Etymology

The etymology needs to be included.174.3.125.23 (talk) 06:56, 10 November 2014 (UTC)

Fixed in the current revision 24.61.244.65 (talk) 16:40, 2 February 2022 (UTC)

Inconsistent with degree matrix

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. 109.144.196.64 (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)

Fixed in the current revision 24.61.244.65 (talk) 02:12, 1 February 2022 (UTC)

The name of the random-walk normalized Laplacian

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)

Fixed in the current revision 24.61.244.65 (talk) 02:12, 1 February 2022 (UTC)

Definition of random-walk normalized Laplacian

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 $D^{-1}A$ is non negative). It seems that it should be defined as $I-D^{-1}A$, however, the identity involving $L_{sym}$, is correct according to the current definition. — Preceding unsigned comment added by 151.100.102.21 (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 $v_{i}$ to any other adjacent node is uniform, so $1/deg(v_{i})$ and definitely not negative! — Preceding unsigned comment added by 146.107.107.116 (talk) 08:49, 9 September 2016 (UTC)

Fixed in the current revision 24.61.244.65 (talk) 02:12, 1 February 2022 (UTC)

Assessment comment

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)

Fixed in the current revision 24.61.244.65 (talk) 01:58, 1 February 2022 (UTC)

Unreliable reference

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)

Fixed in the current revision 24.61.244.65 (talk) 02:09, 1 February 2022 (UTC)

Edge weights

I don't see anything about edge weights. To my naïve eyes, the talk about multigraphs comes closest. Would someone who knows what they are doing add to that section or a new section a discussion about how undirected and directed graphs with edge weights are handled? Thank you 66.194.253.20 (talk) 13:57, 27 January 2021 (UTC)

Fixed in the current revision 24.61.244.65 (talk) 01:58, 1 February 2022 (UTC)

Section "Open source software implementations"

This section only list three links to general software (Julia and SciPy), which are not dedicated Laplacian matrices nor graphs, except perhaps NetworkX. Shouldn't it be removed? The link to NetworkX could be moved to the following section ("Application software"). Gapato (talk) 10:04, 14 June 2024 (UTC)