Abstract
In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a
graph and the relationship of these functions to random walks on the graph.
We commence by discussing the set of eigenfunctions supported at the vertices,
and demonstrate the relationship of these eigenfunctions to the classical
random walk on the graph. Then, from an analysis of functions supported
only on the interior of edges, we develop a method for explicitly calculating
the edge-interior eigenfunctions of the edge-based Laplacian. This reveals
a connection between the edge-based Laplacian and the adjacency matrix
of backtrackless random walk on the graph. The edge-based eigenfunctions
therefore correspond to some eigenfunctions of the normalised Hashimoto
matrix.
graph and the relationship of these functions to random walks on the graph.
We commence by discussing the set of eigenfunctions supported at the vertices,
and demonstrate the relationship of these eigenfunctions to the classical
random walk on the graph. Then, from an analysis of functions supported
only on the interior of edges, we develop a method for explicitly calculating
the edge-interior eigenfunctions of the edge-based Laplacian. This reveals
a connection between the edge-based Laplacian and the adjacency matrix
of backtrackless random walk on the graph. The edge-based eigenfunctions
therefore correspond to some eigenfunctions of the normalised Hashimoto
matrix.
| Original language | English |
|---|---|
| Pages (from-to) | 4183-4189 |
| Number of pages | 7 |
| Journal | Linear Algebra and its applications |
| Volume | 438 |
| Issue number | 11 |
| Publication status | Published - 14 Feb 2013 |
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