On bipartite graphs of defect at most 4

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and diameter D. In this context, the Moore bipartite bound Mb({\Delta},D) represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite graphs, have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq 2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is, bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3. However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite $(\Delta,d,-4)$-graphs; the complete catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon \leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with {\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq 3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure

On graphs of defect at most 2

In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D). Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called Moore graphs, turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is, ({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect. Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1, ({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is 2D. Second, and most important, we prove the non-existence of ({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome, together with a proof on the non-existence of (4, 3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second census of ({\Delta},D,-2)-graphs known at present, the first being the one of (3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other results of this paper include necessary conditions for the existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5 such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure

An overview of the degree/diameter problem for directed, undirected and mixed graphs

A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.Peer Reviewe

Evolution of the Potential Energy Surface with Size for Lennard-Jones Clusters

Disconnectivity graphs are used to characterize the potential energy surfaces of Lennard-Jones clusters containing 13, 19, 31, 38, 55 and 75 atoms. This set includes members which exhibit either one or two `funnels' whose low-energy regions may be dominated by a single deep minimum or contain a number of competing structures. The graphs evolve in size due to these specific size effects and an exponential increase in the number of local minima with the number of atoms. To combat the vast number of minima we investigate the use of monotonic sequence basins as the fundamental topographical unit. Finally, we examine disconnectivity graphs for a transformed energy landscape to explain why the transformation provides a useful approach to the global optimization problem.Comment: 13 pages, 8 figures, revte

Surface areas of equifacetal polytopes inscribed in the unit sphere S2\mathbb{S}^2

This article is concerned with the problem of placing seven or eight points on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ so that the surface area of the convex hull of the points is maximized. In each case, the solution is given for convex hulls with congruent isosceles or congruent equilateral triangular facets.Comment: 13 pages, 1 table, 15 figure

The MOLDY short-range molecular dynamics package

We describe a parallelised version of the MOLDY molecular dynamics program. This Fortran code is aimed at systems which may be described by short-range potentials and specifically those which may be addressed with the embedded atom method. This includes a wide range of transition metals and alloys. MOLDY provides a range of options in terms of the molecular dynamics ensemble used and the boundary conditions which may be applied. A number of standard potentials are provided, and the modular structure of the code allows new potentials to be added easily. The code is parallelised using OpenMP and can therefore be run on shared memory systems, including modern multicore processors. Particular attention is paid to the updates required in the main force loop, where synchronisation is often required in OpenMP implementations of molecular dynamics. We examine the performance of the parallel code in detail and give some examples of applications to realistic problems, including the dynamic compression of copper and carbon migration in an iron-carbon alloy