For every integer k 2 and graph G; consider the following natural procedure: if G has a component G0 that is not k-connected, remove G0 if jG0j k, otherwise remove a cutset U V (G0) with jU j < k; do the same with the remaining graph until only k-connected components are left or all vertices are removed. We are interested when this procedure stops after removing o (jGj) vertices. Surprisingly, ...

For any given graph and an integer k, the number of connected components with k vertices in the graph is investigated. For the vertex set of size n and the maximum degree , the number is bounded above by (e ) k ( 1)k . The factor k is essential, since we give the lower bound n 2 k 1 for k < 2n . We also show that all connected components with k vertices are enumerable in O n+m+ (e ) k 1 k 2 log...

We provide an approach to distribute the calculation of PageRank, by splitting the graph into its strongly connected components. As we prove, the global ranking may be calculated componentwise, as long as the rankings of pages directly linking to the current component are already known. Depending on the structure of the WWW, this approach approach may be used to calculate the ranking on several...

LetX be a smooth projective geometrically connected curve over a finite field with function field K. Let G be a connected semisimple group scheme over X . Under certain hypothesis we prove the equality of two numbers associated with G. The first is an arithmetic invariant, its Tamagawa number. The second, is a geometric invariant, the number of connected components of the moduli stack of G-tors...

The problem of finding k-edge-connected components is a fundamental problem in computer science. Given a graph G = (V, E), the problem is to partition the vertex set V into {V1, V2,…, Vh}, where each Vi is maximized, such that for any two vertices x and y in Vi, there are k edge-disjoint paths connecting them. In this paper, we present an algorithm to solve this problem for all k. The algorithm...

Self-organizing maps have been adopted in many fields as the data visualization method of choice. The unified distance matrix is the de facto standard for evaluating and interpreting self-organizing maps. In large, high-dimensional problems clusters can be difficult to identify in the plain unified distance matrix. Here we introduce an enhanced version of the unified distance matrix in which cl...

Graphics Processing Units (GPU) are application specific accelerators which provide high performance to cost ratio and are widely available and used, hence places them as a ubiquitous accelerator. A computing paradigm based on the same is the general purpose computing on the GPU (GPGPU) model. The GPU due to its graphics lineage is better suited for the data-parallel, data-regular algorithms. T...

The Graph Isomorphism problem has both theoretical and practical interest. In this paper we present an algorithm, called conauto-1.2, that efficiently tests whether two graphs are isomorphic, and finds an isomorphism if they are. This algorithm is an improved version of the algorithm conauto, which has been shown to be very fast for random graphs and several families of hard graphs [9]. In this...

When controlling a complex networked system it is not feasible to control the full network because many networks, including biological, technological, and social systems, are massive in size and complexity. But neither is it necessary to control the full network. In complex networks, the giant connected components provide the essential information about the entire system. How to control these g...