We classify the distance-regular Cayley graphs with least eigenvalue $$-2$$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the min-cut problem. Our new semidefinite programming relaxation of the min-cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. Fixing results in several smaller subproblems that need to be solved to obtain the new bound. To efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we develop a heuristic approach based on the well-known reverse Cuthill–McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs under consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices.

The present issue of Designs, Codes and Cryptography is devoted to the theme "Geometric and Algebraic Combinatorics".A central concept in this research area is the Association Scheme.On one hand it can be a tool for a better understanding of combinatorial objects, such as error correcting codes, block designs, point-line incidence geometries, and permutation groups.On the other hand, many association schemes are interesting objects in themselves.This includes the strongly regular and distance-regular graphs.Algebraic tools like eigenvalues are extremely important for studying association schemes, but are also useful tools in their own right for studying the structure of graphs.Also incidence geometries, especially projective and affine geometries over finite fields are often related to association schemes, but as expected, here geometric methods play a more important role.The issue contains fourteen articles, which we'll briefly review.

In many fields, we come across problems where we want to optimize several conflicting objectives simultaneously. To find a good solution for such multiobjective optimization problems, an approximation of the Pareto set is often generated. In this paper, we consider the approximation of higher-dimensional convex Pareto sets using sandwich algorithms. We extend higher-dimensional sandwich algorithms in three different ways. First, we introduce the new concept of adding dummy points to the inner approximation of a Pareto set. By using these dummy points, we can determine accurate inner and outer approximations more efficiently, i.e., using less time-consuming optimizations. Second, we introduce a new method for the calculation of an error measure that is easy to interpret. Third, we show how transforming certain objective functions can improve the results of sandwich algorithms and extend their applicability to certain nonconvex problems. To show the effect of these enhancements, we make a numerical comparison using four test cases, including a four-dimensional case from the field of intensity-modulated radiation therapy. The results of the different cases show that we can achieve an accurate approximation using significantly fewer optimizations by using the enhancements.

The design of computer experiments is an important step in black-box evaluation and optimization processes. When dealing with multiple black-box functions the need often arises to construct designs for all black boxes jointly, instead of individually. These so-called nested designs are particularly useful as training and test sets for fitting and validating metamodels, respectively. Furthermore, nested designs can be used to deal with linking parameters and sequential evaluations. In this paper, we introduce one-dimensional nested maximin designs. We show how to nest two designs optimally and develop a heuristic to nest three and four designs. These nested maximin designs can be downloaded from the website http://www.spacefillingdesigns.nl . Furthermore, it is proven that the loss in space-fillingness, with respect to traditional maximin designs, is at most 14.64 and 19.21%, when nesting two and three designs, respectively.