Hermitian Adjacency Matrix

• The Hermitian adjacency matrix is a square matrix that uses complex entries (1, i, -i) to represent undirected, directed, and mixed graph connections.
• Its Hermitian nature guarantees real eigenvalues, facilitating the use of spectral methods to analyze graph connectivity, cycle structures, and cospectrality through switching operations.
• This matrix framework is applied in non-backtracking walks, quantum walks, and community detection, advancing both theoretical research and practical network analysis.

The Hermitian adjacency matrix is an essential concept in the study of graphs, particularly within the fields of linear algebra and graph theory. It is used to represent a complex graph in a matrix form where each edge and its orientation—whether directed, undirected, or mixed—determines the corresponding entry's value using complex numbers. This matrix plays a crucial role in encoding both the structure and orientation of graphs, facilitating eigenvalue analysis and enabling the exploration of graph properties from an algebraic perspective.

1. Definition and Construction

A Hermitian adjacency matrix $H(G)$ of a graph $G$ is a square matrix where the vertices of the graph correspond to both the rows and columns. The entries $h_{ij}$ depend on the type of edge connecting vertex $i$ to vertex $j$:

• Undirected edges (bidirectional): If the edge is undirected, the entry is $1$.
• Directed edges: If there is a directed edge from vertex $i$ to vertex $j$, the entry is $i$ (the imaginary unit), and for a reverse direction from $j$ to $i$, the entry is $-i$.
• No connection or disjoint vertices: The entry is $0$.

This matrix is Hermitian, meaning $H(G) = [H(G)]^\ast$, ensuring that all eigenvalues are real, a characteristic shared with conventional symmetric adjacency matrices used in undirected graphs.

2. Spectral Properties and Implications

The Hermitian property of the adjacency matrix gives it several advantageous spectral properties. Its eigenvalues are real, enabling the use of spectral methods to evaluate graph properties, such as connectivity and cycle structures. The interlacing theorem is applicable, allowing insights into the behavior and properties of graph substructures.

The spectral radius, defined as the largest absolute value of the eigenvalues, is particularly insightful for understanding graph dynamics. There are bounds on the spectral radius, linking it to the graph's maximum degree, thus providing a combinatorial interpretation of spectral properties.

3. Applications in Graph Theory

The Hermitian adjacency matrix is adept for examining mixed graphs—those containing both undirected and directed edges—offering a tool to generalize concepts from undirected graphs. It aids in characterizing graph cospectrality, meaning different graphs sharing the same spectrum, which is essential for graph isomorphism testing and spectral graph theory.

Cospectrality and Switching Operations

Switching operations, like two-way and three-way switchings, can transform a graph into another cospectral one. This is significant for understanding how changes to edge orientations affect the graph spectrum, and in classifying graphs into switching equivalence classes, thereby revealing deeper symmetries.

4. Role in Non-backtracking Walks

The Hermitian adjacency matrix integrates with non-backtracking matrices, which avoid revisiting vertices immediately, offering enhancements in community detection in complex networks. By encoding orientations and strengths of edges, the complex non-backtracking matrix extends its applicability to directed graphs, emphasizing cluster structures particularly in sparse conditions.

5. Extensions to Other Roots of Unity

Research has extended Hermitian adjacency matrices to incorporate other unitary complex numbers (roots of unity) for entries besides $i$. These adaptations can yield matrices like the N-matrix, using complex numbers such as sixth roots of unity, offering flexibility in representing graph relationships, maintaining the Hermitic property, and opening new avenues of spectral analysis on a broader spectrum of graph types.

6. Applications in Quantum Walks

Generalized Hermitian adjacency matrices find use in defining quantum walks on graphs, where the matrix entries serve as transition amplitudes. These quantum walks use the matrix to evolve according to quantum mechanical rules, providing models that are sensitive to graph topology, phase, and directionality, signaling applications in areas like quantum algorithms and quantum computing.

Conclusion

The Hermitian adjacency matrix serves as a critical tool in the representation and analysis of complex, directed, and mixed graphs, with applications spanning from theoretical graph properties to practical implementations in network analysis and quantum computing. Its algebraic formulation allows a unification of structural and directional graph properties, lending itself to a wide range of spectral methods applicable across traditional and cutting-edge mathematical contexts. The continuous development of these matrices fosters deeper linkages between graph theory, linear algebra, and fields leveraging complex networks.