Eigenfunctions of the Edge-Based Laplacian on a Graph

TL;DR

This paper investigates the eigenfunctions of the edge-based Laplacian on graphs, revealing their connections to random walks and the adjacency matrix, and provides methods for explicit calculation of these eigenfunctions.

Contribution

It introduces a method to explicitly compute edge-interior eigenfunctions and uncovers their relationship with backtrackless random walks and the Hashimoto matrix.

Findings

Eigenfunctions supported at vertices relate to classical random walks.

Explicit calculation method for edge-interior eigenfunctions is developed.

Edge-based Laplacian eigenfunctions connect to the Hashimoto matrix eigenfunctions.

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.