Spectral Characteristics of Graphs and Matrices

The spectral characteristics of graphs and matrices form a central pillar of modern mathematical physics and network theory. At its core, the adjacency matrix of a graph encodes connectivity by mapping vertices to matrix indices, while the Laplacian matrix captures diffusion and flow through its degree–difference structure. The eigenvalues and eigenvectors of these matrices reveal fundamental properties such as connectivity, community structure, synchronisation dynamics and resilience to perturbations. Beyond classical graphs, extensions to signed and gain graphs introduce phase or sign information on edges, enriching spectral signatures and modelling phenomena from power grids to quantum networks. Simultaneously, the study of characteristic and permanent polynomials of matrices provides a combinatorial gateway to determinants, nullity and rank profiles, underpinning both theoretical advances and algorithmic implementations. Recent methodological strides include the deployment of interlacing families of polynomials to bound spectral radii and the development of combinatorial decompositions that express characteristic polynomials via induced substructures. These advances have catalysed progress in designing expander and Ramanujan graphs, optimising spectral clustering and elucidating phase coherence in complex networks. Practical applications span from rapid community detection in social and biological systems to stability analysis in engineered infrastructures and insights into molecular vibrations in chemistry. Such spectral tools continue to bridge discrete and continuous domains, offering a versatile framework for probing the interplay between structure and dynamics across large‐scale systems.

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Spectral Characteristics of Graphs and Matrices publication trend

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Technical terms

Adjacency matrix: A square matrix whose entries indicate the presence or absence of edges between pairs of vertices in a graph.

Laplacian matrix: A matrix defined as the difference between the degree matrix and the adjacency matrix, used to study diffusion and flow properties in networks.

Hermitian adjacency matrix: An extension of the adjacency matrix for directed graphs, where entries are complex conjugate pairs assigned to oriented edges.

Spectral radius: The largest absolute value of the eigenvalues of a matrix, reflecting maximal connectivity or dynamic response.

Characteristic polynomial: A polynomial whose roots are the eigenvalues of a matrix, encoding spectral properties in its coefficients.

Gain graph: A graph in which each edge is labelled by a group element or scalar phase, generalising signed graphs through multiplicative edge weights.

Interlacing family: A set of polynomial sequences whose roots satisfy nested inequalities, enabling bounds on eigenvalues through combinatorial constructions.

References

1. On characteristic and permanent polynomials of a matrix. Special Matrices (2017).
2. Interlacing families and the Hermitian spectral norm of digraphs. Linear Algebra and its Applications (2019).
3. Eigenvalues of complex unit gain graphs and gain regularity. Special Matrices (2024).