Oriented Gain Graphs, Line Graphs and Eigenvalues

TL;DR

This paper develops a theory of orientation for gain graphs, explores their line graphs, and establishes matrix properties for complex unit gain graphs, linking incidence and adjacency matrices.

Contribution

It introduces a new orientation framework for gain graphs and analyzes their line graphs, extending classical graph concepts to gain graphs with complex units.

Findings

Established matrix properties for complex unit gain graphs.

Linked incidence matrix of gain graphs to line graph adjacency matrix.

Generalized orientation concept for gain graphs.

Abstract

A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.