Wildcard dimensions, coding theory and fault-tolerant meshes and hypercubes

Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First we prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. We then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes we construct d-dimensional meshes and tori with wildcard dimensions. Finally, we show how these constructions can be used to tolerate edge and node faults in mesh and torus networks

T-colorings, divisibility and the circular chromatic number

LetTbe a $T-set$, i.e., a finite set of nonnegative integers satisfying $0∈T$,and$Gbe$ a graph. In the paper we study relations between theT-edge spansesp $T(G)$ and $espd⊙T(G$), wheredis a positive integer and $d⊙T={0≤t≤d(maxT+ 1) :d|t⇒t/d∈T}$.We show that $espd⊙T(G)$ =$despT(G)−r$, wherer, $0≤r≤d−1$, is aninteger that depends onTandG. Next we focus on the $caseT={0}$ and show that $espd⊙{0}(G) =⌈d(χc(G)−1)⌉$,where $χc(G)$ is the circular chromatic number ofG. This result allows us toformulate several interesting conclusions that include a new formula for thecircular chromatic $numberχc(G)$ = $1 + inf{espd⊙{0}(G)/d:d≥1}$ $2R$ and a proof that the formula for the $T-edge$ span of powers of cycles, statedas conjecture in [Y. Zhao, W. He and R. Cao,The edge span ofT-coloringon graphCdn, Appl. Math. Lett. 19 (2006) 647–651], is true

Some Properties on Estrada Index of Folded Hypercubes Networks

Let G be a simple graph with n vertices and let λ1,λ2,…,λn be the eigenvalues of its adjacency matrix; the Estrada index EEG of the graph G is defined as the sum of the terms eλi,  i=1,2,…,n. The n-dimensional folded hypercube networks FQn are an important and attractive variant of the n-dimensional hypercube networks Qn, which are obtained from Qn by adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networks FQn by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networks FQn are proposed

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed

Partial multinode broadcast and partial exchange algorithms for d-dimensional meshes

Caption title. "Revision of January 1992."Includes bibliographical references (p. 24-26).Supported by NSF. NSF-ECS-8519058 Supported by ARO. DAAL03-86-K-0171by Emmanouel A. Varvarigos and Dimitri P. Bertsekas

A Modified TOPSIS Method Based on D

Multicriteria decision-making (MCDM) is an important branch of operations research which composes multiple-criteria to make decision. TOPSIS is an effective method in handling MCDM problem, while there still exist some shortcomings about it. Upon facing the MCDM problem, various types of uncertainty are inevitable such as incompleteness, fuzziness, and imprecision result from the powerlessness of human beings subjective judgment. However, the TOPSIS method cannot adequately deal with these types of uncertainties. In this paper, a D-TOPSIS method is proposed for MCDM problem based on a new effective and feasible representation of uncertain information, called D numbers. The D-TOPSIS method is an extension of the classical TOPSIS method. Within the proposed method, D numbers theory denotes the decision matrix given by experts considering the interrelation of multicriteria. An application about human resources selection, which essentially is a multicriteria decision-making problem, is conducted to demonstrate the effectiveness of the proposed D-TOPSIS method