Cycle Spaces of Digraphs

TL;DR

This paper explores the cycle space of directed graphs, providing a combinatorial basis description and linking it to quantum walk transition matrices.

Contribution

It introduces a combinatorial basis for the cycle space of digraphs and connects it to quantum walk transition matrices.

Findings

Provided a basis description for the cycle space of digraphs.

Established a connection between cycle spaces and quantum walk transition matrices.

Extended classical graph cycle space concepts to directed graphs.

Abstract

The cycle space of a graph corresponds to the kernel of an incidence matrix. We investigate an analogous subspace for digraphs. In the case of digraphs of graphs, where every edge is replaced by two oppositely directed arcs, we give a combinatorial description of a basis of such a space. We are motivated by a connection to the transition matrices of discrete-time quantum walks.