Eigenvalues of Cayley graphs

DOI10.37236/8569zbMath1492.05093arXiv1809.09829MaRDI QIDQ2138565

Xiaogang Liu, Sanming Zhou

Publication date: 12 May 2022

Published in: The Electronic Journal of Combinatorics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1809.09829

zbMATH Keywords

eigenvaluesCayley Graphs

Mathematics Subject Classification ID

Graphs and abstract algebra (groups, rings, fields, etc.) (05C25) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50)

Related Items

Integral Cayley graphs over a certain nonabelian group ⋮ Quantum symmetries of Cayley graphs of abelian groups ⋮ Algebraic degrees of 2-Cayley digraphs over abelian groups ⋮ State transfer on integral mixed circulant graphs ⋮ Splitting fields of mixed Cayley graphs over abelian groups ⋮ Algebraic degree of Cayley graphs over dicyclic and semi-dihedral groups ⋮ Quantum state transfer on integral oriented circulant graphs ⋮ One-matching bi-Cayley graphs and homogeneous bi-Cayley graphs over finite cyclic groups ⋮ Perfect state transfer on quasi-abelian semi-Cayley graphs ⋮ On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2 ⋮ The integrality of distance spectra of quasiabelian 2-Cayley graphs ⋮ Generalized Paley graphs equienergetic with their complements ⋮ Splitting fields of some matrices of normal (mixed) Cayley graphs ⋮ Aldous' spectral gap property for normal Cayley graphs on symmetric groups ⋮ Integral Cayley graphs over dicyclic group ⋮ Energies of complements of unitary one-matching bi-Cayley graphs over commutative rings

Uses Software

• GAP

Cites Work

• Closed walks and eigenvalues of abelian Cayley graphs
• Cayley properties of merged Johnson graphs
• Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3
• Ramanujan graphs on cosets of \(\operatorname{PGL}_2(\mathbb F_q)\)
• Longest induced cycles in circulant graphs
• Spectral estimates for abelian Cayley graphs
• A note on the Ádám conjecture for double loops.
• Pretty good state transfer on circulant graphs
• Note on exact values of multiplicities of eigenvalues of the Star graph
• Groups all of whose undirected Cayley graphs are integral
• Isoperimetric numbers of graphs
• Pair state transfer
• General properties of some families of graphs defined by systems of equations
• Unsolved Problems in Group Theory. The Kourovka Notebook
• On the graph of nilpotent matrix group of length one
• A Few Remarks on the Octopus Inequality and Aldous’ Spectral Gap Conjecture
• On finite groups all of whose cubic Cayley graphs are integral
• CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS
• Groups all of whose undirected Cayley graphs are determined by their spectra
• Discrepancy and eigenvalues of Cayley graphs
• Graph Energy
• Para-Krawtchouk polynomials on a bi-lattice and a quantum spin chain with perfect state transfer
• Quartic integral Cayley graphs
• RAMANUJAN CAYLEY GRAPHS OF FROBENIUS GROUPS
• The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs
• Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension
• Lamplighter groups, de Brujin graphs, spider-web graphs and their spectra
• On the Cayley Graphs of Brandt Semigroups
• Expander graphs in pure and applied mathematics
• CONVOLUTIONS OF RAMANUJAN SUMS AND INTEGRAL CIRCULANT GRAPHS
• When can perfect state transfer occur?
• Perfect state transfer, integral circulants and join of graphs
• Distance spectrum of graph compositions
• On the spectrum of unitary finite-Euclidean graphs
• Integral Cayley Graphs and Groups
• A method to determine algebraically integral Cayley digraphs on finite Abelian group
• ON THE EIGENVALUES OF N-CAYLEY GRAPHS: A SURVEY
• Eigenvalues and automorphisms of a graph
• Group representations that resist random sampling
• Erdős–Ko–Rado Theorems: Algebraic Approaches
• Expander graphs and their applications
• Quasirandom Groups
• A Class of Arc-Transitive Cayley Graphs as Models for Interconnection Networks
• Proof of Aldous’ spectral gap conjecture
• Eigenspaces of Hamming graphs and unitary Cayley graphs
• PERFECT, EFFICIENT, STATE TRANSFER AND ITS APPLICATION AS A CONSTRUCTIVE TOOL
• Arc transitive covering digraphs and their eigenvalues
• A Survey on Spectra of infinite Graphs
• Diameters and Eigenvalues
• Generating a random permutation with random transpositions
• How Good is the Goemans--Williamson MAX CUT Algorithm?
• Random Cayley graphs and expanders
• Algebraic Graph Theory
• On the bisection width of the transposition network
• Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
• Finite Euclidean graphs over rings
• A solution of the isomorphism problem for circulant graphs
• Perfect state transfer on gcd-graphs
• Quasirandom Cayley graphs
• Character sums, automorphic forms, equidistribution, and Ramanujan graphs. Part II. Eigenvalues of Terras
• CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER
• Unnamed Item
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• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Unnamed Item
• On the eigenvalues of normal edge-transitive Cayley graphs
• Distance-regular graphs
• The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups
• A class of \textit{gcd}-graphs having perfect state transfer
• Integral Cayley sum graphs and groups
• Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups
• Integral Cayley multigraphs over abelian and Hamiltonian groups
• Integral circulant Ramanujan graphs of prime power order
• On the characteristic polynomial of \(n\)-Cayley digraphs
• Algebraic Cayley graphs over finite fields
• On the spectrum of Wenger graphs
• Toroidal fullerenes with the Cayley graph structures
• Resistance distances and the Kirchhoff index in Cayley graphs
• Skew-adjacency matrices of graphs
• Spectrum of Cayley graphs on the symmetric group generated by transpositions
• New results on the energy of integral circulant graphs
• Automorphism groups of rational circulant graphs
• Integral sets and Cayley graphs of finite groups
• The maximal energy of classes of integral circulant graphs
• Extremal energies of integral circulant graphs via multiplicativity
• Diameters, distortion, and eigenvalues
• A counterexample to the Alon-Saks-Seymour conjecture and related problems
• Color energy of a unitary Cayley graph
• Cayley sum color and anti-circulant graphs
• On the unitary Cayley graphs of matrix algebras
• On groups all of whose undirected Cayley graphs of bounded valency are integral
• Which weighted circulant networks have perfect state transfer?
• A note on the spectrum of linearized Wenger graphs
• A large family of cospectral Cayley graphs over dihedral groups
• Recursive cubes of rings as models for interconnection networks
• Unimodular graphs and Eisenstein sums
• The distance spectra of Cayley graphs of Coxeter groups
• Further results on the perfect state transfer in integral circulant graphs
• On the automorphism group of integral circulant graphs
• Integral quartic Cayley graphs on Abelian groups
• Integral Cayley graphs defined by greatest common divisors
• Perfect state transfer in cubelike graphs
• Energy of unitary Cayley graphs and gcd-graphs
• On the skew energy of orientations of hypercubes
• How does Sudoku acquire integral eigenvalues?
• On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
• Gaussian integral circulant digraphs
• Periodic graphs
• More counterexamples to the Alon-Saks-Seymour and rank-coloring conjectures
• Representation theory of finite groups. An introductory approach.
• Some results for the periodicity and perfect state transfer
• Spectra of graphs
• State transfer on graphs
• Subspace arrangements and property T.
• Spectral and combinatorial properties of some algebraically defined graphs
• Laplace eigenvalues of graphs---a survey
• Integral circulant graphs of prime power order with maximal energy
• The energy of unitary Cayley graphs
• Some classes of integral circulant graphs either allowing or not allowing perfect state transfer
• Spectra of Cayley graphs of complex reflection groups
• Largest independent sets of certain regular subgraphs of the derangement graph
• On distance-regular Cayley graphs on abelian groups
• Setwise intersecting families of permutations
• Partial difference sets
• \(\lambda_ 1\), isoperimetric inequalities for graphs, and superconcentrators
• Geometric bounds for eigenvalues of Markov chains
• Integral circulant graphs
• Spectral analysis of the affine graph over the finite ring
• On the energy of some circulant graphs
• Perfect state transfer in integral circulant graphs
• Isospectral Cayley graphs of some finite simple groups
• Distance-regular Cayley graphs on dihedral groups
• The spectrum of Platonic graphs over finite fields
• Intersecting families in the alternating group and direct product of symmetric groups
• On strongly regular bicirculants
• New bounds for the \(\max\)-\(k\)-cut and chromatic number of a graph
• Cayley-type graphs for group-subgroup pairs
• Some relations between power graphs and Cayley graphs
• The smallest eigenvalues of the 1-point fixing graph
• Spectra of digraphs
• Eigenvalues of the derangement graph
• The spectrum of semi-Cayley graphs over abelian groups
• Perfect state transfer in integral circulant graphs of non-square-free order
• A classification of Ramanujan unitary Cayley graphs
• On restricted unitary Cayley graphs and symplectic transformations modulo \(n\)
• Integral Cayley graphs over Abelian groups
• Distance spectra and distance energy of integral circulant graphs
• Large sets in finite fields are sumsets
• Cayley sum graphs and eigenvalues of \((3,6)\)-fullerenes
• On the spectrum of the derangement graph
• Developments on spectral characterizations of graphs
• Strongly regular tri-Cayley graphs
• On generalised Paley graphs and their automorphism groups
• Cayley digraphs with normal adjacency matrices
• Connected graphs as subgraphs of Cayley graphs: conditions on hamiltonicity
• Finite Euclidean graphs and Ramanujan graphs
• On the rational spectra of graphs with Abelian Singer groups
• Random shuffles and group representations
• Explicit construction of linear sized tolerant networks
• Ramanujan graphs
• Eigenvalues and expanders
• Cayley graphs of finite groups
• The group and the minimal polynomial of a graph
• Rational G-matrices with rational eigenvalues
• Eigenvalues of graphs and digraphs
• Survey of Spectra of Laplacians on Finite Symmetric Spaces
• Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks
• A survey on integral graphs
• COMPUTING THE EIGENVALUES OF CAYLEY GRAPHS OF ORDER p2q
• Perfect state transfer on Cayley graphs over dihedral groups
• On the spectrum of Cayley graphs related to the finite groups
• On the flip graphs on perfect matchings of complete graphs and signed reversal graphs
• On the spectrum of finite Cayley graphs
• Perfect state transfer in NEPS of some graphs
• Aldous’s spectral gap conjecture for normal sets
• Unitary homogeneous bi-Cayley graphs over finite commutative rings
• A CLASSIFICATION OF FINITE GROUPS WITH INTEGRAL BI-CAYLEY GRAPHS
• Open problems in the spectral theory of signed graphs
• Perfect state transfer in unitary Cayley graphs and gcd-graphs
• Distance-regular Cayley graphs with small valency
• Integral Cayley Graphs over Finite Groups
• On the energy and Estrada index of Cayley graphs
• On So's conjecture for integral circulant graphs
• PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS
• On groups admitting no integral Cayley graphs besides complete multipartite graphs
• The energy of integral circulant graphs with prime power order
• Some Properties of a Cayley Graph of a Commutative Ring
• GCD-graphs and NEPS of complete graphs
• BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS
• Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales
• The exact maximal energy of integral circulant graphs with prime power order
• Strongly regular m-Cayley circulant graphs and digraphs
• Point-symmetric graphs with a prime number of points
• On the p-rank of the design matrix of a difference set
• Isomorphism problem for a special class of graphs
• Graphs with circulant adjacency matrices
• On the Addressing Problem for Loop Switching
• On a class of arithmetical convolutions
• Energy and Laplacian energy of unitary addition Cayley graphs
• PRIME BASES OF WEAKLY PRIME SUBMODULES AND THE WEAK RADICAL OF SUBMODULES
• Intersecting families of permutations
• On Cayley graphs on the symmetric group generated by transpositions
• Maximal energy graphs
• Circulant digraphs determined by their spectra
• On the mixed adjacency matrix of a mixed graph
• On Cayley sum graphs of non-abelian groups
• The gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear
• Strongly regular semi-Cayley graphs
• Spectra of Cayley graphs
• Spectra of graphs with transitive groups
• Strong regularity and circulant graphs
• On the spectra of certain graphs arising from finite fields
• The distance-regular graphs of valency four
• Partial difference triples
• A survey of partial difference sets
• Minimal eigenvalue of the Coxeter Laplacian for the symmetric group
• Rate of convergence for shuffling cards by transpositions
• The spectral gap of the ferromagnetic \(XXZ\) chain
• On orbital regular graphs and Frobenius graphs
• Distance-regular circulants
• Which graphs are determined by their spectrum?
• Spectra of Heisenberg graphs over finite rings
• Finite analogues of non-Euclidean spaces and Ramanujan graphs.
• The energy of a graph
• On the spectral Ádám property for circulant graphs
• Integral Cayley graphs over dihedral groups
• The spectrum of eigenvalues for certain subgraphs of the \(k\)-point fixing graph
• More circulant graphs exhibiting pretty good state transfer
• Pretty good quantum state transfer in symmetric spin networks via magnetic field
• Perfect state transfer on abelian Cayley graphs
• Distance powers of unitary Cayley graphs
• Energies of unitary one-matching bi-Cayley graphs over finite commutative rings
• Integral Cayley graphs over dicyclic group
• Construction of strongly regular Cayley graphs based on three-valued Gauss periods
• A new infinite family of hemisystems of the Hermitian surface
• Diameter bounds and recursive properties of Full-Flag Johnson graphs
• On the change of the Jordan form under the transition from the adjacency matrix of a vertex-transitive digraph to its principal submatrix of co-order one
• The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters
• On cycles in the sequence of unitary Cayley graphs
• Some graphs with small second eigenvalue
• Symmetric polynomials and the center of the symmetric group ring
• Sparse quasi-random graphs
• On isomorphisms of finite Cayley graphs---a survey
• Non-Ramanujancy of Euclidean graphs of order 2\(^{r}\)
• Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem
• Tight estimates for eigenvalues of regular graphs
• Eigenvalue bounds and girths of graphs of finite, upper half-planes
• Graph spectra for finite upper half planes over rings
• Finite upper half planes over finite fields
• Counting pure \(k\)-cycles in sequences of Cayley graphs
• Finite analogues of Euclidean space
• Characterization of quantum circulant networks having perfect state transfer
• Circulant digraphs integral over number fields
• Spectral properties of unitary Cayley graphs of finite commutative rings
• Distance powers and distance matrices of integral Cayley graphs over abelian groups
• Integral Cayley graphs generated by distance sets in vector spaces over finite fields
• The second eigenvalue of some normal Cayley graphs of highly transitive groups
• Addressing graph products and distance-regular graphs
• Integral and distance integral Cayley graphs over generalized dihedral groups
• Perfect state transfer on weighted abelian Cayley graphs
• Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups
• Perfect state transfer on Cayley graphs over dihedral groups: the non-normal case
• Unitary Cayley graphs of matrix rings over finite commutative rings
• Fractional revival and association schemes
• Perfect state transfer in NEPS of complete graphs
• Integral equienergetic non-isospectral unitary Cayley graphs
• A large family of cospectral Cayley graphs over dicyclic groups
• Spectra and eigenspaces of arbitrary lifts of graphs
• Cayley graph on symmetric group generated by elements fixing \(k\) points
• Remarks on singular Cayley graphs and vanishing elements of simple groups
• The spectra of lifted digraphs
• Pretty good state transfer on Cayley graphs over dihedral groups
• \(PI\)-eigenfunctions of the Star graphs
• On the spectral gap of some Cayley graphs on the Weyl group \(W(B_n)\)
• Integral Cayley graphs
• The second largest eigenvalues of some Cayley graphs on alternating groups
• On a relationship between Cayley graphs and \(G\)-graphs with some applications
• Integral circulant Ramanujan graphs via multiplicativity and ultrafriable integers
• Linearized Wenger graphs
• Perfect state transfer on distance-regular graphs and association schemes
• Quadratic unitary Cayley graphs of finite commutative rings
• On the partition associated to the smallest eigenvalues of the \(k\)-point fixing graph
• Integral pentavalent Cayley graphs on abelian or dihedral groups
• On integral Cayley sum graphs
• The spectral gap of graphs arising from substring reversals
• Distance-regular Cayley graphs with least eigenvalue \(-2\)
• On the eigenvalues of the Coxeter Laplacian.
• On certain integral Schreier graphs of the symmetric group
• Some properties of unitary Cayley graphs
• On the energy of unitary Cayley graphs
• Sudoku graphs are integral
• Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap
• On the unitary Cayley graph of a finite ring
• Which Cayley graphs are integral?
• Uniform expansion bounds for Cayley graphs of \(\text{SL}_2(\mathbb F_p)\).
• The eigenvalues of the graphs \(D(4,q)\)
• Uniform mixing on Cayley graphs
• On the spectrum of Cayley graphs
• Multiplicities of eigenvalues of the star graph
• Enumeration of cubic Cayley graphs on dihedral groups
• Algebraic Cayley graphs over finite local rings
• On the eigenvalues of certain Cayley graphs and arrangement graphs
• Integer invariants of abelian Cayley graphs
• Solving the Ku-Wales conjecture on the eigenvalues of the derangement graph
• On the average energy of circulant graphs
• On the extreme eigenvalues of regular graphs.