147 research outputs found
On the strong partition dimension of graphs
We present a different way to obtain generators of metric spaces having the
property that the ``position'' of every element of the space is uniquely
determined by the distances from the elements of the generators. Specifically
we introduce a generator based on a partition of the metric space into sets of
elements. The sets of the partition will work as the new elements which will
uniquely determine the position of each single element of the space. A set
of vertices of a connected graph strongly resolves two different vertices
if either or
, where . An ordered vertex partition of
a graph is a strong resolving partition for if every two different
vertices of belonging to the same set of the partition are strongly
resolved by some set of . A strong resolving partition of minimum
cardinality is called a strong partition basis and its cardinality the strong
partition dimension. In this article we introduce the concepts of strong
resolving partition and strong partition dimension and we begin with the study
of its mathematical properties. We give some realizability results for this
parameter and we also obtain tight bounds and closed formulae for the strong
metric dimension of several graphs.Comment: 16 page
The Wiener polarity index of benzenoid systems and nanotubes
In this paper, we consider a molecular descriptor called the Wiener polarity
index, which is defined as the number of unordered pairs of vertices at
distance three in a graph. Molecular descriptors play a fundamental role in
chemistry, materials engineering, and in drug design since they can be
correlated with a large number of physico-chemical properties of molecules. As
the main result, we develop a method for computing the Wiener polarity index
for two basic and most commonly studied families of molecular graphs, benzenoid
systems and carbon nanotubes. The obtained method is then used to find a closed
formula for the Wiener polarity index of any benzenoid system. Moreover, we
also compute this index for zig-zag and armchair nanotubes
Phase transitions of extremal cuts for the configuration model
The -section width and the Max-Cut for the configuration model are shown
to exhibit phase transitions according to the values of certain parameters of
the asymptotic degree distribution. These transitions mirror those observed on
Erd\H{o}s-R\'enyi random graphs, established by Luczak and McDiarmid (2001),
and Coppersmith et al. (2004), respectively
A modular network treatment of Baars' Global Workspace consciousness model
Network theory provides an alternative to the renormalization and phase transition methods used in Wallace's (2005a) treatment of Baars' Global Workspace model. Like the earlier study, the new analysis produces the workplace itself, the tunable threshold of consciousness, and the essential role for embedding contexts, in an explicitly analytic 'necessary conditions' manner which suffers neither the mereological fallacy inherent to brain-only theories nor the sufficiency indeterminacy of neural network or agent-based simulations. This suggests that the new approach, and the earlier, represent different analytically solvable limits in a broad continuum of possible models, analogous to the differences between bond and site percolation or between the two and many-body limits of classical mechanics. The development significantly extends the theoretical foundations for an empirical general cognitive model (GCM) based on the Shannon-McMillan Theorem. Patterned after the general linear model which reflects the Central Limit Theorem, the proposed technique should be both useful for the reduction of expermiental data on consciousness and in the design of devices with capacities which may transcend those of conventional machines and provide new perspectives on the varieties of biological consciousness
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