Given a graph with an adjacency matrix $\bf A$, powers of this matrix give the number of walks from vertices. That is, $({\bf A}^k)_{ij}$ gives the number of walks from nodes $i$ to $j$ in $k$ steps. Is there any nice physical iterpertation for the powers of the Laplacian matrix?
Powers of the Laplacian matrix
5
$\begingroup$
linear-algebra
graph-theory
1 Answers
6
In my opinion, it is not natural to take powers of the Laplacian in isolation. That is, the Laplacian should not be thought of as a linear operator that one is interested in iterating. There are two points of view that I know of about what the Laplacian "is":
- It's the matrix representation of the quadratic form $\sum_{(v, w) \in E} (f(v) - f(w))^2$ where $f$ is a function on the vertices. Matrices of quadratic forms are not meant to be composed.
- It generates a one-dimensional Lie algebra acting on the space of functions on the vertices. This perspective is relevant to the answer I gave to this math.SE question, where by using the Laplacian to define certain differential equations it becomes natural to exponentiate, rather than take powers of, the Laplacian. The powers appear in the exponential, but indirectly.
-
0@Hooked: sure, but I'm not sure this gets you anything with as direct an interpretation as the powers of the adjacency matrix. – 2011-02-11