Article
Physics, Mathematical
Pierre-Antoine Bernard, Nicolas Crampe, Luc Vinet
Summary: This paper studies free fermions on Johnson graphs J(n, k) and calculates the entanglement entropy of sets of neighborhoods. For a subsystem composed of a single neighborhood, an analytical expression is provided through the decomposition in irreducible submodules of the Terwilliger algebra of J(n, k) embedded in two copies of su(2). For a subsystem composed of multiple neighborhoods, the construction of a block-tridiagonal operator that commutes with the entanglement Hamiltonian is presented, highlighting its usefulness in computing the entropy, and discussing the area law pre-factor.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Physics, Particles & Fields
Pierre-Antoine Bernard, Nicolas Crampe, Luc Vinet
Summary: This article focuses on free fermions on Hamming graphs H(d, q) and computes the entanglement entropy for two types of subsystems. An analytical expression is obtained for subsets of vertices that form Hamming subgraphs. For subsets corresponding to a neighborhood, a decomposition in irreducible submodules of the Terwilliger algebra of H(d, q) also yields a closed formula for the entanglement entropy. The article also shows how to construct a block-tridiagonal operator for subsystems made out of multiple neighborhoods, which commutes with the entanglement Hamiltonian and is identified as a BC-Gaudin magnet Hamiltonian in a magnetic field and is diagonalized by the modified algebraic Bethe ansatz.
Article
Mathematics
Mark S. MacLean, Safet Penjic
Summary: The paper discusses bipartite distance-regular graphs with diameter at least 4 and valency at least 3, constructing combinatorially-defined spanning sets for T-modules of endpoint 2 with certain assumptions, and examining the action of the adjacency matrix on these sets. By using this T-module, combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 are constructed for various graph examples. Additionally, a list of several other graphs satisfying the conditions is provided.
DISCRETE MATHEMATICS
(2021)
Article
Mathematics
Blas Fernandez
Summary: The main result of this paper is a combinatorial characterization of a certain property of a graph Gamma, which has a unique irreducible T-module with endpoint 1, and that this T-module is thin.
JOURNAL OF ALGEBRAIC COMBINATORICS
(2022)
Article
Mathematics
Blas Fernandez, Stefko Miklavic
Summary: This paper investigates the Terwilliger algebra of a finite, simple, connected, and bipartite graph, based on a specific vertex, and derives the property of the unique reducible T-module in the graph, providing a combinatorial characterization of this property.
EUROPEAN JOURNAL OF COMBINATORICS
(2021)
Article
Mathematics
Akihide Hanaki, Masayoshi Yoshikawa
Summary: For a finite connected simple graph, the Terwilliger algebra is a matrix algebra generated by the adjacency matrix and idempotents corresponding to the distance partition with respect to a fixed vertex. We will consider algebras defined by two other partitions and the centralizer algebra of the stabilizer of the fixed vertex in the automorphism group of the graph. We will give some methods to compute such algebras and examples for various graphs.
DISCRETE MATHEMATICS
(2023)
Article
Mathematics
Kazumasa Nomura, Paul Terwilliger
Summary: This paper explores Leonard pairs with spin, discussing the characteristics of spin models in finite-dimensional vector spaces and their application in algebras.
JOURNAL OF COMBINATORIAL THEORY SERIES A
(2021)
Article
Mathematics
Pierre-Antoine Bernard, Nicolas Crampe, Luc Vinet
Summary: It is demonstrated that the adjacency matrix of a symplectic dual polar graph, restricted to the eigenspaces of an abelian automorphism subgroup, can serve as the adjacency matrix of a weighted subspace lattice. This connection is then utilized to determine the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs, and the multiplicities of the isomorphic submodules are provided.
DISCRETE MATHEMATICS
(2022)
Article
Mathematics
Jack H. Koolen, Jae-Ho Lee, Ying-Ying Tan
Summary: This paper discusses the pseudo-vertex transitivity of distance-regular graphs with diameters 2, 3, and 4. The results show that a strongly regular graph is pseudo-vertex transitive if and only if its local graphs have the same spectrum. Additionally, it is shown that Taylor graphs and antipodal tight graphs are pseudo-vertex transitive.
DISCRETE MATHEMATICS
(2022)
Article
Mathematics
John Vincent S. Morales, Tessie M. Palma
Summary: The study focuses on the Doob graph D = D(n, m) formed by n copies of the Shrikhande graph and m copies of the complete graph K-4. It introduces the Terwilliger algebra and quantum decomposition of D with respect to a fixed vertex x. The paper also discusses the quantum adjacency algebra of the graph and demonstrates the action of the special orthogonal Lie algebra on the standard module for D, proving that it is generated by the center and the homomorphic image of the universal enveloping algebra U(so(4)).
JOURNAL OF ALGEBRAIC COMBINATORICS
(2021)
Article
Mathematics, Applied
Ebrahim Ghorbani, Masoumeh Koohestani
Summary: Gu, Jost, Liu, and Stadler (2016) introduced the concept of spectral limit for sequences of graphs and determined the spectral limit for sequences of strongly-regular and distance-regular graphs.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics, Applied
Monther rashed Alfuraidan
Summary: We provide a complete description of strongly regular graphs with a distance-transitive group of automorphisms. This is the first time that the complete list is presented in one place, although parts of it have been mentioned in previous literature. The description is complemented with discussions on the corresponding distance-transitive groups and additional properties of the graphs. An open problem is also pointed out.
CARPATHIAN JOURNAL OF MATHEMATICS
(2022)
Article
Computer Science, Theory & Methods
Pin-Chieh Tseng, Ching-Yi Lai, Wei-Hsuan Yu
Summary: In this study, we propose more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver's semidefinite programming bounds on A(n, d). In particular, we improve the semidefinite programming bounds on A(18, 4) to 6551.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Mathematics, Applied
Hiroaki Taniguchi
Summary: This note proves that the incidence graphs of the semibiplanes constructed from dimensional dual hyperovals are distance regular graphs if the dual hyperovals are doubly dual hyperovals (DDHOs), extending the result in [12].
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics, Applied
Nicholas W. Mayers, Nicholas Russoniello
Summary: A (2k + 1)-dimensional Lie algebra is called contact if it satisfies the condition ϕ∧(dϕ)k = 0, where ϕ is a one-form. This paper extends recent research and presents a combinatorial procedure for generating contact, type-A Lie poset algebras with chains of arbitrary cardinality in their associated posets. The authors conjecture that their construction provides a complete characterization.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)