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

TL;DR

The paper develops the integrated adjacency matrix $\mathcal{I}(G)$ for mixed graphs, proving it uniquely encodes the graph and sharing its spectrum with an associated graph $G^A$ via $A(G^A)=\mathcal{I}(G)$. It links spectral information to combinatorial structures through alternating walks and mixed components, and derives exact spectra for several structured graphs, a determinant formula, and general eigenvalue bounds. By establishing a bijection between mixed components of $G$ and components of $G^A$, the work unifies the spectral theory of mixed graphs with classical graph spectra. The results lay a foundation for further spectral matrices (integrated Laplacian, signless Laplacian, etc.) to be explored in Part II.

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.

Figures (7)

• Figure 1: A mixed graph $G$ along with its undirected part $G_u$ and directed part $G_d$.
• Figure 2: The associated graph $G^A$ of the mixed graph $G$ shown in Figure \ref{['fig-mixed-exmpl']}
• Figure 3: A mixed graph $G$ and it's special submixed graphs $G_1$, $G_2$, $G_3$, $G_4$, $G_5$ and $G_6$
• Figure 4: An uniconnected mixed graph $G$
• Figure 5: The mixed graph $G$ and it's mixed component $H$ having the AP property

Theorems & Definitions (85)

• Example 3.1
• Definition 3.1
• Example 3.2
• Proposition 4.1
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof