New matrices for the spectral theory of mixed graphs, part I

TL;DR

This paper introduces the integrated adjacency matrix for mixed graphs, linking spectral properties to graph structure and establishing relationships between eigenvalues and specific mixed graph configurations.

Contribution

It presents a novel matrix for mixed graphs and connects spectral analysis to structural graph properties, expanding spectral graph theory.

Findings

The integrated adjacency matrix uniquely determines a mixed graph.

Spectral analysis of the matrix reveals structural properties of mixed graphs.

Relationships between eigenvalues and mixed graph structures are established.

Abstract

In this paper, we introduce a matrix for a mixed graph, called the integrated adjacency matrix. This matrix uniquely determines a mixed graph, as long as the indices of the matrix are specified. Additionally, we associate an (undirected) graph with each mixed graph, enabling the spectral analysis of the integrated adjacency matrix to connect the structural properties of the mixed graph and its associated graph. Furthermore, we define certain mixed graph structures and establish their relationships to the eigenvalues of the integrated adjacency matrix.