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 almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called $m$-walk-regularity. Another studied concept is that of $m$-partial distance-regularity or, informally, distance-regularity up to distance $m$. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of $(\ell,m)$-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem

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

A characterization of Q-polynomial association schemes

We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.Comment: 8 pages, no figur

On Almost Distance-Regular Graphs

2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial