# On the adjacency matrix of a complex unit gain graph

## TL;DR

This paper investigates the spectral properties of adjacency matrices in complex unit gain graphs, establishing bounds, characterizations, and conditions relating eigenvalues to the underlying graph structure.

## Contribution

It introduces new bounds and spectral characterizations for complex unit gain graphs, linking eigenvalues to graph bipartiteness and gain conditions.

## Key findings

- Eigenvalue bounds for complex unit gain graphs
- Characterization of bipartite graphs via eigenvalues
- Conditions for eigenvalues of gain graphs to match underlying graphs

## Abstract

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron-Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same.