526 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Switching Reconstruction of Digraphs
Switching about a vertex in a digraph means to reverse the direction of every
edge incident with that vertex. Bondy and Mercier introduced the problem of
whether a digraph can be reconstructed up to isomorphism from the multiset of
isomorphism types of digraphs obtained by switching about each vertex. Since
the largest known non-reconstructible oriented graphs have 8 vertices, it is
natural to ask whether there are any larger non-reconstructible graphs. In this
paper we continue the investigation of this question. We find that there are
exactly 44 non-reconstructible oriented graphs whose underlying undirected
graphs have maximum degree at most 2. We also determine the full set of
switching-stable oriented graphs, which are those graphs for which all
switchings return a digraph isomorphic to the original
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Some Investigations in the Theory of Graphs
Not availabl
On palimpsests in neural memory: an information theory viewpoint
The finite capacity of neural memory and the
reconsolidation phenomenon suggest it is important to be able
to update stored information as in a palimpsest, where new
information overwrites old information. Moreover, changing
information in memory is metabolically costly. In this paper, we
suggest that information-theoretic approaches may inform the
fundamental limits in constructing such a memory system. In
particular, we define malleable coding, that considers not only
representation length but also ease of representation update,
thereby encouraging some form of recycling to convert an old
codeword into a new one. Malleability cost is the difficulty of
synchronizing compressed versions, and malleable codes are of
particular interest when representing information and modifying
the representation are both expensive. We examine the tradeoff
between compression efficiency and malleability cost, under a
malleability metric defined with respect to a string edit distance.
This introduces a metric topology to the compressed domain. We
characterize the exact set of achievable rates and malleability as
the solution of a subgraph isomorphism problem. This is all done
within the optimization approach to biology framework.Accepted manuscrip
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