Simple random walk on distance-regular graphs

A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field

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

On the diameter of an ideal

We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal

Hypercubes, Leonard triples and the anticommutator spin algebra

This paper is about three classes of objects: Leonard triples, distance-regular graphs and the modules for the anticommutator spin algebra. Let \K denote an algebraically closed field of characteristic zero. Let $V$ denote a vector space over \K with finite positive dimension. A Leonard triple on $V$ is an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis for $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. The Leonard triples of interest to us are said to be totally B/AB and of Bannai/Ito type. Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the anticommutator spin algebra $\mathcal{A}$, the unital associative \K-algebra defined by generators $x,y,z$ and relations$$xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.$$ Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter $D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp. $\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp. $\tilde{Q}_{D}$). We obtain the following. When $D$ is even (resp. odd), we show that there exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp. $\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency matrices respectively. We classify the resulting irreducible $\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show that actions of the adjacency, dual adjacency and weighted adjacency matrices for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp. $\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.Comment: arXiv admin note: text overlap with arXiv:0705.0518 by other author