In this paper we study variants of the widely used spectral clustering that partitions a graph into k clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix, and (2) grouping the embedded points into k clusters via k-means algorithms. We show that, for a wide class of graphs, spectral clustering gives a good approximation of the optimal clustering. While this approach was proposed in the early 1990s and has comprehensive applications, prior to our work similar results were known only for graphs generated from stochastic models.We also give a nearly-linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures.

We present a 2-local quantum algorithm for graph isomorphism GI based on an adiabatic protocol. By exploiting continuous-time quantum-walks, we are able to avoid a mere diffusion over all possible configurations and to significantly reduce the dimensionality of the visited space. Within this restricted space, the graph isomorphism problem can be translated into the search of a satisfying assignment to a 2-SAT formula without resorting to perturbation gadgets or projective techniques. We present an analysis of the execution time of the algorithm on small instances of the graph isomorphism problem and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.

Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of algorithmic design methods for graph clustering. Most of these methods, however, are based on complicated spectral techniques or convex optimisation, and cannot be directly applied for clustering many networks that occur in practice, whose information is often collected on different sites. Designing a simple and distributed clustering algorithm is of great interest, and has wide applications for processing big datasets.In this paper we present a simple and distributed algorithm for graph clustering: for a wide class of graphs that are characterised by a strong cluster-structure, our algorithm finishes in a poly-logarithmic number of rounds, and recovers a partition of the graph close to optimal. One of the main components behind our algorithm is a sampling scheme that, given a dense graph as input, produces a sparse subgraph that provably preserves the clusterstructure of the input. Compared with previous sparsification algorithms that require Laplacian solvers or involve combinatorial constructions, this component is easy to implement in a distributed way and runs fast in practice.

A convenient, simple, and high-yielding five-step synthesis of a sphingosine acceptor from phytosphingosine is reported, and its behavior in glycosylation reactions is described. Different synthetic paths to sphingosine acceptors using tetrachlorophthalimide as a protecting group for the sphingosine amino function and different glycosylation methods have been explored. Among the acceptors tested, the easiest accessible acceptor, unprotected on the two hydroxyl groups in positions 1 and 3, was regioselectively glycosylated on the primary position, the regioselectivity depending on the donor used.

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