Talk:Spectral graph theory

Characteristic polynomial is a graph invariant (depending on the combinatorial structure of the graph), but not a topological invariant (depending only on the topological space associated to the graph). For example, subdividing an edge with extra vertices does not change the topological type, but does change the (degree of) the char poly.

Are results like these also to be added here? :

the absolute value of an eigenvalue is at most the maximal degree

a regular graph has the degree of each vertex as eigenvalue exactly once iff the graph is connected

J, the matrix that is one everywhere, is in the adjacency algebra, if and only if, the graph is regular and connected

if the graph is connected, the diameter is at most the number of distinct eigenvalues minus oneEvilbu 20:34, 6 February 2006 (UTC)

It is a quite good point to talk about the topological invariant regarding the characteristic polynomial of a graph.

Also it would be better if this article had more editorial work. For instance, its first paragraph just seems to be cluttered with facts. More separation would lead to a better quality exposition. What do others think?

Rogério Brito 07:56, 20 December 2006 (UTC)