Spectral Properties of Cayley Graphs
Summary
Cayley graphs provide a natural bridge between algebra and combinatorics by encoding a group and a generating set as vertices and edges, respectively. The eigenvalues of their adjacency and Laplacian matrices form the graph spectrum, revealing deep information about symmetry, connectivity and expansion. High degrees of regularity in Cayley graphs permit explicit diagonalisation in many cases, while bounds on eigenvalue magnitudes underpin the theory of expander families. In particular, Ramanujan Cayley graphs achieve optimal spectral gaps, making them exceptional for rapid mixing of random walks and robust communication networks. Spectral measures such as algebraic connectivity quantify resilience to vertex removal and synchronisation properties in consensus processes. Beyond pure mathematics, spectral analysis of Cayley graphs finds application in designing cryptographic primitives, error-correcting codes and quantum walks, illustrating a rich interplay between group-theoretic structure and global network dynamics.
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Spectral Properties of Cayley Graphs publication trend
The graph below shows the total number of articles in spectral properties of cayley graphs across all publications each year (not limited to Nature Index journals).
Technical terms
Cayley graph: A graph whose vertices are group elements and edges correspond to multiplication by a fixed generating set.
Spectrum: The multiset of eigenvalues of a graph’s adjacency or Laplacian matrix, reflecting connectivity and expansion.
Spectral gap: The difference between the largest and next-largest eigenvalue (in absolute value) of the adjacency matrix, or between the first two Laplacian eigenvalues.
Algebraic connectivity: The second-smallest eigenvalue of the Laplacian, measuring how well a graph remains connected under vertex removal.
Ramanujan graph: A regular graph whose non-trivial eigenvalues lie within the theoretical bound given by the Alon–Boppana theorem, optimising expansion.
References
- Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4n. Mathematics (2019).
- The Square of Some Generalized Hamming Graphs. Mathematics (2023).
- Spectra and eigenspaces of arbitrary lifts of graphs. Journal of Algebraic Combinatorics (2021).
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