Decomposition of complete graphs into 5-cubes

Necessary conditions for the complete graph on n vertices to have a decomposition into 5-cubes are that 5 divides it - 1 and 80 divides it (it - 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for it even, thus completing the spectrum problem for the 5-cube and lending further weight to a long-standing conjecture of Kotzig. (c) 2005 Wiley Periodicals, Inc.

Some equitably 2-colourable cycle decompositions of multipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has x(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar x(i) - x(j)vertical bar

On Hamilton cycle decomposition of 6-regular circulant graphs

The circulant graph Sn, where S ⊆ Zn \ {0}, has vertex set Zn and edge set {{x, x + s}|x ∈ Zn, s ∈ S}. It is shown that there is a Hamilton cycle decomposition of every 6-regular circulant graph Sn in which S has an element of order n.

Coloring random graphs and maximizing local diversity

We study a variation of the graph coloring problem on random graphs of finite average connectivity. Given the number of colors, we aim to maximize the number of different colors at neighboring vertices (i.e. one edge distance) of any vertex. Two efficient algorithms, belief propagation and Walksat are adapted to carry out this task. We present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution. The problem and the suggested algorithms have practical relevance since various applications, such as distributed storage, can be mapped onto this problem.

Resource allocation in sparse graphs

Resource allocation in sparsely connected networks, a representative problem of systems with real variables, is studied using the replica and Bethe approximation methods. An efficient distributed algorithm is devised on the basis of insights gained from the analysis and is examined using numerical simulations,showing excellent performance and full agreement with the theoretical results. The physical properties of the resource allocation model are discussed.

Message passing for task redistribution on sparse graphs

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.

Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach. © 2007 The American Physical Society.

Inference and optimization of real edges on sparse graphs:A statistical physics perspective

Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real edge variables that interact at the network nodes are obtained in various cases. When applied to the representative problem of network resource allocation, efficient distributed algorithms are also devised. Scaling properties with respect to the network connectivity and the resource availability are found, and links to probabilistic Bayesian approximation methods are established. Different cost measures are considered and algorithmic solutions in the various cases are devised and examined numerically. Simulation results are in full agreement with the theory. © 2007 The American Physical Society.

Investigation into visual defects attributed to Vigabatrin

Vigabatrin (VGB) is a transaminase inhibitor that elicits its anitepileptic effect by increasing GABA concentrations in the brain and retina. - Assess whether certain factors predispose patients to develop severe visual field loss. - Develop a sensitive algorithm for investigating the progression of visual field loss. - Determine the most sensitive clinical regimen for diagnosing VGB-attributed visual field loss. - Investigate whether the reports of central retinal sparing are accurate. The investigations have resulted in a number of significant findings: - The anatomical evidence in combination with the pattern of visual field loss suggests that the damage induced by VGB therapy occurs at retinal level, and is most likely a toxic effect. - The quantitative algorithm, designed within the course of this investigation, provided increased sensitivity in determining the severity of visual field loss. - Maximum VGB dose predisposes patients to develop severe visual field loss. - The SITA Standard algorithm was found to be as sensitive and significantly faster, in diagnosing visual field defects attributed to VGB, when compared to the Full Threshold algorithm. The Full Threshold was found to be the most repeatable between visits. - The normal SWAP 10-2 database provided an effective method of differentiating SWAP defects. - SWAP, FDT and the mfERG have increased sensitivity in detecting visual field loss attributed to VGB. The pattern of visual field loss from these investigations suggests that VGB produces a diffuse effect across the retina including subtle central abnormalities and more severe peripheral defects. - Abnormalities detected using the mfERG have suggested that VGB adversely affects the photoreceptors Müller, amacrine and ganglion cells in the retina. An urgent review of the manufacturers recommended maximum dose for VGB is required.

Competition for shortest paths on sparse graphs

Optimal paths connecting randomly selected network nodes and fixed routers are studied analytically in the presence of a nonlinear overlap cost that penalizes congestion. Routing becomes more difficult as the number of selected nodes increases and exhibits ergodicity breaking in the case of multiple routers. The ground state of such systems reveals nonmonotonic complex behaviors in average path length and algorithmic convergence, depending on the network topology, and densities of communicating nodes and routers. A distributed linearly scalable routing algorithm is also devised. © 2012 American Physical Society.

Research allocation on sparse graphs

DUE TO COPYRIGHT RESTRICTIONS ONLY AVAILABLE FOR CONSULTATION AT ASTON UNIVERSITY LIBRARY AND INFORMATION SERVICES WITH PRIOR ARRANGEMENT

Algorithms for solving large random graphs

DUE TO COPYRIGHT RESTRICTIONS ONLY AVAILABLE FOR CONSULTATION AT ASTON UNIVERSITY LIBRARY AND INFORMATION SERVICES WITH PRIOR ARRANGEMENT

Belief propagation on dense graphs with few strong links

DUE TO COPYRIGHT RESTRICTIONS ONLY AVAILABLE FOR CONSULTATION AT ASTON UNIVERSITY LIBRARY AND INFORMATION SERVICES WITH PRIOR ARRANGEMENT