Laplacian matrices are important objects in the field of Spectral Graph Theory.
Many properties of the graph can be read off easily from the properties of the corresponding Laplacian Matrix.
For instance:
- The multiplicity of the eigenvalue zero gives the number of connected components of the graph.
- The largest eigenvalue is 2 if and only if a connected component of the graph is a non-trivial bipartite graph.
There are plenty more results like these.
I would recommend you try the book by Fan Chung (Ronald Graham's wife, I believe) conveniently titled Spectral Graph Theory. Here is a link to the book page: http://www.math.ucsd.edu/~fan/research/revised.html
An explanation of the possible motivation for using the Laplacian (and it's normalized form, which is what you seem to be talking about, when you say norm) appears in the first chapter of that book: http://www.math.ucsd.edu/~fan/research/cb/ch1.pdf
Also search the web for a linear algebra proof of Friendship theorem, which uses matrices. Even though Laplacian matrices are not used, I would still recommend you read that, as it will give you an idea about the power of spectral methods in graph theory.
Hope that helps.
