An algebraic approach to lifts of digraphs

TL;DR

This paper introduces an algebraic framework using quotient-like matrices to analyze lifts of digraphs, linking their structure to spectra, with applications to well-known graphs like Petersen and Hoffman-Singleton.

Contribution

It develops a novel algebraic approach using complex polynomial matrices to study digraph lifts and their spectra, especially for Abelian groups.

Findings

Complete characterization of spectra for a new family of digraphs including generalized Petersen graphs.

Spectral analysis of the Hoffman-Singleton graph.

Introduction of a quotient-like matrix that captures the structure of digraph lifts.

Abstract

We study the relationship between two key concepts in the theory of (di)graphs: the quotient digraph, and the lift $\Gamma^{\alpha}$ of a base (voltage) digraph. These techniques contract or expand a given digraph in order to study its characteristics, or obtain more involved structures. This study is carried out by introducing a quotient-like matrix, with complex polynomial entries, which fully represents $\Gamma^{\alpha}$. In particular, such a matrix gives the quotient matrix of a regular partition of $\Gamma^{\alpha}$, and when the involved group is Abelian, it completely determines the spectrum of $\Gamma^{\alpha}$. As some examples of our techniques, we study some basic properties of the Alegre digraph. In addition we completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and that of the Hoffman-Singleton graph.