Spectral properties of weighted line digraphs

TL;DR

This paper investigates the spectral properties of weighted line digraphs derived from connected undirected graphs, revealing how boundary operators influence their spectra and applying findings to analyze the spectrum of the Grover matrix's positive support.

Contribution

It introduces a framework connecting boundary operators to the spectra of weighted line digraphs and applies this to spectral analysis of quantum walk operators.

Findings

Spectrum of the positive support of cube of Grover matrix determined

Weighted line digraphs' spectra depend on boundary operators and Hilbert spaces

Application to large girth graphs enhances understanding of quantum walk spectra

Abstract

In this paper, we treat some weighted line digraphs which are induced by a connected and undirected graph. For a given graph $G$, the adjacency matrix of the weighted line digraph $W$ is determined by a boundary operator from an arc-based space to a vertex-based space. We see that depending on the boundary operator and the Hilbert spaces, $W$ has different kind of an underlying stochastic transition operator. As an application, we obtain the spectrum of the positive support of cube of the Grover matrix in a large girth of the graph.