Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Taut distance-regular graphs and the subconstituent algebra

We consider a bipartite distance-regular graph $G$ with diameter $D$ at least 4 and valency $k$ at least 3. We obtain upper and lower bounds for the local eigenvalues of $G$ in terms of the intersection numbers of $G$ and the eigenvalues of $G$. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of those thin irreducible $T$-modules that have endpoint 2 and dimension $D-3$. In an earlier paper the first author defined what it means for $G$ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the thin irreducible $T$-modules mentioned above.Comment: 29 page

A bilinear form relating two Leonard systems

Let $\Phi$, $\Phi'$ be Leonard systems over a field $\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $\Phi$, $\Phi'$, respectively. In this paper, we introduce and discuss a balanced bilinear form on $V\times V'$. Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs. We characterize a balanced bilinear form from several points of view.Comment: 15 page

Investigation of continuous-time quantum walk on root lattice AnA_n and honeycomb lattice

The continuous-time quantum walk (CTQW) on root lattice $A_n$ (known as hexagonal lattice for $n=2$) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group $Z^{\otimes n}_m$ and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice $A_n$ and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice $A_n$ and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite graphs is approximated with finite ones. Also it is shown that the Bose-Mesner algebras of our constructed association schemes (called $n$-variable $P$-polynomial) can be generated by $n$ commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of $n$-variable orthogonal polynomials which are special cases of \textit{generalized} Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the suppersymmetric structure of finite honeycomb lattices is revealed. Keywords: underlying graphs of association schemes, continuous-time quantum walk, orthogonal polynomials, spectral distribution. PACs Index: 03.65.UdComment: 41 pages, 4 figure

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page