1,271 research outputs found
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
A multipath analysis of biswapped networks.
Biswapped networks of the form have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of vertex-disjoint paths joining any two distinct vertices in , where is the connectivity of . In doing so, we obtain an upper bound of on the -diameter of , where is the diameter of and the -diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network that finds mutually vertex-disjoint paths in joining any distinct vertices and does this in time polynomial in , and (and independently of the number of vertices of ). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network that finds mutually vertex-disjoint paths joining any distinct vertices in so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in , and . We also show that if is Hamiltonian then is Hamiltonian, and that if is a Cayley graph then is a Cayley graph
From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses
Given discrete degrees of freedom (spins) on a graph interacting via an
energy function, what can be said about the energy local minima and associated
inherent structures? Using the lid algorithm in the context of a spin glass
energy function, we investigate the properties of the energy landscape for a
variety of graph topologies. First, we find that the multiplicity Ns of the
inherent structures generically has a lognormal distribution. In addition, the
large volume limit of ln/ differs from unity, except for the
Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the
growth of the height of the energy barrier between the two degenerate ground
states and the size of the associated valleys. For finite connectivity models,
changing the topology of the underlying graph does not modify qualitatively the
energy landscape, but at the quantitative level the models can differ
substantially.Comment: 10 pages, 9 figs, slightly improved presentation, more references,
accepted for publication in Phys Rev
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