128 research outputs found
Which finitely generated Abelian groups admit isomorphic Cayley graphs?
We show that Cayley graphs of finitely generated Abelian groups are rather
rigid. As a consequence we obtain that two finitely generated Abelian groups
admit isomorphic Cayley graphs if and only if they have the same rank and their
torsion parts have the same cardinality. The proof uses only elementary
arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3:
small corrections; to appear in Geometriae Dedicat
On graphs of defect at most 2
In this paper we consider the degree/diameter problem, namely, given natural
numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of
vertices in a graph of maximum degree {\Delta} and diameter D. In this context,
the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D).
Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called
Moore graphs, turned out to be very rare. Therefore, it is very interesting to
investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and
order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is,
({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect.
Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1,
({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper
focuses on graphs of defect 2. Building on the approaches developed in [11] we
obtain several new important results on this family of graphs. First, we prove
that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is
2D. Second, and most important, we prove the non-existence of
({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome,
together with a proof on the non-existence of (4, 3,-2)-graphs (also provided
in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs
with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second
census of ({\Delta},D,-2)-graphs known at present, the first being the one of
(3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other
results of this paper include necessary conditions for the existence of
({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the
non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5
such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure
Minimal chordal sense of direction and circulant graphs
A sense of direction is an edge labeling on graphs that follows a globally
consistent scheme and is known to considerably reduce the complexity of several
distributed problems. In this paper, we study a particular instance of sense of
direction, called a chordal sense of direction (CSD). In special, we identify
the class of k-regular graphs that admit a CSD with exactly k labels (a minimal
CSD). We prove that connected graphs in this class are Hamiltonian and that the
class is equivalent to that of circulant graphs, presenting an efficient
(polynomial-time) way of recognizing it when the graphs' degree k is fixed
Boolean networks synchronism sensitivity and XOR circulant networks convergence time
In this paper are presented first results of a theoretical study on the role
of non-monotone interactions in Boolean automata networks. We propose to
analyse the contribution of non-monotony to the diversity and complexity in
their dynamical behaviours according to two axes. The first one consists in
supporting the idea that non-monotony has a peculiar influence on the
sensitivity to synchronism of such networks. It leads us to the second axis
that presents preliminary results and builds an understanding of the dynamical
behaviours, in particular concerning convergence times, of specific
non-monotone Boolean automata networks called XOR circulant networks.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
On the complexity of acyclic modules in automata networks
Modules were introduced as an extension of Boolean automata networks. They
have inputs which are used in the computation said modules perform, and can be
used to wire modules with each other. In the present paper we extend this new
formalism and study the specific case of acyclic modules. These modules prove
to be well described in their limit behavior by functions called output
functions. We provide other results that offer an upper bound on the number of
attractors in an acyclic module when wired recursively into an automata
network, alongside a diversity of complexity results around the difficulty of
deciding the existence of cycles depending on the number of inputs and the size
of said cycle.Comment: 21 page
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