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

Krein parameters and antipodal tight graphs with diameter 3 and 4

AbstractWe determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q114 vanishes

Three-dimensional entanglement: knots, knits and nets

Three-dimensional entanglement, including knots, periodic arrays of woven filaments (weavings) and periodic arrays of interpenetrating networks (nets), forms an integral part of the analysis of structure within the natural sciences. This thesis constructs a catalogue of 3-periodic entanglements via a scaffold of Triply-Periodic Minimal Surfaces (TPMS). The two-dimensional Hyperbolic plane can be wrapped over a TPMS in much the same way as the two-dimensional Euclidean plane can be wrapped over a cylinder. Thus vertices and edges of free tilings of the Hyperbolic plane, which are tilings by tiles of infinite size, can be wrapped over a TPMS to represent vertices and edges of an array in three-dimensional Euclidean space. In doing this, we harness the simplicity of a two-dimensional surface as compared with 3D space to build our catalogue. We numerically tighten these entangled flexible knits and nets to an ideal conformation that minimises the ratio of edge (or filament) length to diameter. To enable the tightening of periodic entanglements which may contain vertices, we extend the Shrink-On-No-Overlaps algorithm, a simple and fast algorithm for tightening finite knots and links. The ideal geometry of 3-periodic weavings found through the tightening process exposes an interesting physical property: Dilatancy. The cooperative straightening of the filaments with a fixed diameter induces an expansion of the material accompanied with an increase in the free volume of the material. Further, we predict a dilatant rod packing as the structure of the keratin matrix in the corneocytes of mammalian skin, where the dilatant property of the matrix allows the skin to maintain structural integrity while experiencing a large expansion during the uptake of water

Hypercube-Based Topologies With Incremental Link Redundancy.

Hypercube structures have received a great deal of attention due to the attractive properties inherent to their topology. Parallel algorithms targeted at this topology can be partitioned into many tasks, each of which running on one node processor. A high degree of performance is achievable by running every task individually and concurrently on each node processor available in the hypercube. Nevertheless, the performance can be greatly degraded if the node processors spend much time just communicating with one another. The goal in designing hypercubes is, therefore, to achieve a high ratio of computation time to communication time. The dissertation addresses primarily ways to enhance system performance by minimizing the communication time among processors. The need for improving the performance of hypercube networks is clearly explained. Three novel topologies related to hypercubes with improved performance are proposed and analyzed. Firstly, the Bridged Hypercube (BHC) is introduced. It is shown that this design is remarkably more efficient and cost-effective than the standard hypercube due to its low diameter. Basic routing algorithms such as one to one and broadcasting are developed for the BHC and proven optimal. Shortcomings of the BHC such as its asymmetry and limited application are clearly discussed. The Folded Hypercube (FHC), a symmetric network with low diameter and low degree of the node, is introduced. This new topology is shown to support highly efficient communications among the processors. For the FHC, optimal routing algorithms are developed and proven to be remarkably more efficient than those of the conventional hypercube. For both BHC and FHC, network parameters such as average distance, message traffic density, and communication delay are derived and comparatively analyzed. Lastly, to enhance the fault tolerance of the hypercube, a new design called Fault Tolerant Hypercube (FTH) is proposed. The FTH is shown to exhibit a graceful degradation in performance with the existence of faults. Probabilistic models based on Markov chain are employed to characterize the fault tolerance of the FTH. The results are verified by Monte Carlo simulation. The most attractive feature of all new topologies is the asymptotically zero overhead associated with them. The designs are simple and implementable. These designs can lead themselves to many parallel processing applications requiring high degree of performance