Spectral Properties of Complex Unit Gain Graphs

TL;DR

This paper extends spectral graph theory to complex unit gain graphs by defining key matrices and establishing eigenvalue bounds, enhancing understanding of their spectral properties.

Contribution

It introduces the spectral properties of complex unit gain graphs, including definitions of adjacency, incidence, and Laplacian matrices, and derives eigenvalue bounds.

Findings

Eigenvalue bounds for adjacency matrices

Eigenvalue bounds for Laplacian matrices

Extension of spectral concepts to complex gain graphs

Abstract

A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them. The main results of the paper are eigenvalue bounds for the adjacency and Laplacian matrices.