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whoapao@hotmail.com
science forum beginner

Joined: 22 Mar 2006
Posts: 3

Posted: Sat Jun 24, 2006 9:02 am    Post subject: Laplacian of weighted graphs

Hi,
Got a possibly silly question. Given a graph, the Laplacian is the
degree matrix minus the adjacency matrix. Does one talk about the
Laplacian matrix of a graph where the vertices and edges are labeled,
i.e. each vertex and each edge is assigned some number? Any reference
is appreciated. Thanks in advance.
tchow@lsa.umich.edu

Joined: 15 Sep 2005
Posts: 53

Posted: Sat Jun 24, 2006 5:23 pm    Post subject: Re: Laplacian of weighted graphs

<whoapao@hotmail.com> wrote:
 Quote: Got a possibly silly question. Given a graph, the Laplacian is the degree matrix minus the adjacency matrix. Does one talk about the Laplacian matrix of a graph where the vertices and edges are labeled, i.e. each vertex and each edge is assigned some number? Any reference is appreciated. Thanks in advance.

Weights on the edges are handled in the "obvious" way: the degree of a
vertex is the sum of the weights of the incident edges, and the adjacency
matrix is the weighted adjacency matrix. There isn't going to be a
"reference" for this fact other than the usual references for Laplacians
(Cvetkovic & Doob's "Spectra of Graphs," Biggs's "Algebraic Graph Theory,"
Godsil & Royle's "Algebraic Graph Theory," Chung's "Spectral Graph Theory,"
etc.).

I'm not aware of any variant of the Laplacian that also takes into account
vertex weights.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
Martin Rubey
science forum beginner

Joined: 21 Mar 2005
Posts: 34

Posted: Sun Jun 25, 2006 6:08 am    Post subject: Re: Laplacian of weighted graphs

tchow@lsa.umich.edu writes:

 Quote: I'm not aware of any variant of the Laplacian that also takes into account vertex weights.

There is one by Fan Chung:

www.math.ucsd.edu/~fan/wp/lang.pdf

Martin

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