A characterization of the Hamming graph by strongly closed subgraphs

AbstractThe Hamming graph H(d,q) satisfies the following conditions: (i)For any pair (u,v) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance between u and v. In particular, any strongly closed subgraph is distance-regular.(ii)For any pair (x,y) of vertices at distance d−1 the subgraph induced by the neighbors of y at distance d from x is a clique of size a1+1.In this paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph

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

Variances and Covariances in the Central Limit Theorem for the Output of a Transducer

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-$w$ non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence

Bucolic Complexes

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.Comment: 45 pages, 4 figure