Matching Preclusion and Conditional Matching Preclusion Problems for Twisted Cubes

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find the matching preclusion number and the conditional matching preclusion number for twisted cubes, an improved version of the well-known hypercube. Moreover, we also classify all the optimal matching preclusion sets

Recursive circulants and their embeddings among hypercubes

AbstractWe propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N,d) is defined to be a circulant graph with N nodes and jumps of powers of d. G(N,d) is node symmetric, and has some strong hamiltonian properties. G(N,d) has a recursive structure when N=cdm, 1⩽c<d. We develop a shortest-path routing algorithm in G(cdm,d), and analyze various network metrics of G(cdm,d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2m,4), whose degree is m, compares favorably to the hypercube Qm. G(2m,4) has the maximum possible connectivity, and its diameter is ⌈(3m−1)/4⌉. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in Park and Chwa (Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks ISPAN’94, Kanazawa, Japan, December 1994, pp. 73–80)

Farey Graphs as Models for Complex Networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.Comment: Definitive version published in Theoretical Computer Scienc

Efficient reconfiguration algorithms of de Bruijn and Kautz networks into linear arrays

AbstractIn this paper, we prove the existence of ranking and unranking algorithms on d-ary de Bruijn and Kautz graphs. A ranking algorithm takes as input the label of a node and returns the rank r of that node in a hamiltonian path (0⩽r⩽N−1, where N is the order of the considered graph). An unranking algorithm takes as input an integer r (0⩽r⩽N−1) and returns the label of the rth ranked node in a hamiltonian path. Our results generalize results given by Annexstein for binary de Bruijn graphs. The key of our framework is based on a recursive construction of hamiltonian paths in de Bruijn and Kautz graphs. The construction uses suitable uniform homomorphisms of de Bruijn and Kautz graphs of diameter D on de Bruijn graphs of diameter D−1. Our ranking and unranking algorithms have sequential time complexity in O(D2), where D is the length of node labels

3nj Morphogenesis and Semiclassical Disentangling

Recoupling coefficients (3nj symbols) are unitary transformations between binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum operators. They have been used in a variety of applications in spectroscopy, quantum chemistry and nuclear physics and quite recently also in quantum gravity and quantum computing. These coefficients, naturally associated to cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and analytical features that make them fashinating objects to be studied on their own. In this paper we develop a bottom--up, systematic procedure for the generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and algebraic methods. We provide also a novel approach to the problem of classifying various regimes of semiclassical expansions of 3nj coefficients (asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial, analytical and numerical tools