A new kind of Hermitian matrices for digraphs

TL;DR

This paper introduces a new Hermitian matrix for directed graphs that is more natural than previous versions, aiming to enhance algebraic graph theory analysis.

Contribution

The paper proposes a novel Hermitian matrix for digraphs, improving upon earlier definitions and providing a potentially more natural tool for algebraic graph theory.

Findings

Introduction of a new Hermitian matrix for digraphs

Comparison with previous Hermitian adjacency matrices

Potential applications in algebraic graph theory

Abstract

In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory.