arXiv subjects

Ada Chan

Publications and source records attributed to Ada Chan.

25 records · Page 2Linked to original sources

arXiv2015

Laplacian State Transfer in Coronas

We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of Coutinho and Liu who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends of the work of Fan and Godsil on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.

quant-ph↗

arXiv2013

Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

We study the continuous-time quantum walks on graphs in the adjacency algebra of the $n$-cube and its related distance regular graphs. For $k\geq 2$, we find graphs in the adjacency algebra of $(2^{k+2}-8)$-cube that admit instantaneous uniform mixing at time $π/2^k$ and graphs that have perfect state transfer at time $π/2^k$. We characterize the folded $n$-cubes, the halved $n$-cubes and the folded halved $n$-cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.

math.CO↗

arXiv2011

Hamming graphs in Nomura Algebras

Let A be an association scheme on q\geq 3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n\geq 2, is not the Nomura algebra of a type II matrix. This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices.

math.CO↗

arXiv2007

Type-II Matrices and Combinatorial Structures

Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.

math.CO↗

arXiv2007

Jones Pairs

Motivated by Jones' braid group representations constructed from spin models, we define {\sl a Jones pair} to be a pair of $\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\D_B$ form a representation of a braid group. When $A$ and $B$ are type-II matrices, we call $(A,B)$ {\sl an invertible Jones pair}. We develop the theory of Jones pairs in this thesis. Our aim is to study the connections among association schemes, spin models and four-weight spin models using the viewpoint of Jones pairs. We use Nomura's method to construct a pair of algebras from the matrices $(A,B)$, which we call the Nomura algebras of $(A,B)$. These algebras become the central tool in this thesis. We explore their properties in Chapters \ref{Nomura} and \ref{IINom}. In Chapter \ref{JP}, we introduce Jones pairs. We prove the equivalence of four-weight spin models and invertible Jones pairs. We extend some existing concepts for four-weight spin models to Jones pairs. In Chapter \ref{SpinModels}, we provide new proofs for some well-known results on the Bose-Mesner algebras associated with spin models. We document the main results of the thesis in Chapter \ref{InvJP}. We prove that every four-weight spin model comes from a symmetric spin model (up to odd-gauge equivalence). We present four Bose-Mesner algebras associated to each four-weight spin model. We study the relations among these algebras. In particular, we provide a strategy to search for four-weight spin models. This strategy is analogous to the method given by Bannai, Bannai and Jaeger for finding spin models.

math.CO↗

arXiv2004

Bose-Mesner Algebras attached to Invertible Jones Pairs

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways--by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on $4n$ vertices from each 4-weight spin model on $n$ vertices. In this paper we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyse the structure of the association scheme attached to this spin model.

math.CO↗

arXiv2001

Four-Weight Spin Models and Jones Pairs

We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a ``dual'' pair of association schemes.

math.CO↗