11,756 research outputs found
On the connectivity of p-diamond-free vertex transitive graphs
AbstractLet G be a graph of order n(G), minimum degree δ(G) and connectivity κ(G). We call the graph G maximally connected when κ(G)=δ(G). The graph G is said to be superconnected if every minimum vertex cut isolates a vertex.For an integer p≥1, we define a p-diamond as the graph with p+2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. Usually, the 1-diamond is triangle and the 2-diamond is diamond. We call a graph p-diamond-free if it contains no p-diamond as a (not necessarily induced) subgraph. A graph is vertex transitive if its automorphism group acts transitively on its vertex set.In this paper, we give some sufficient conditions for vertex transitive graphs to be maximally connected. In addition, superconnected p-diamond-free (1≤p≤3) vertex transitive graphs are characterized
On the Fiedler value of large planar graphs
The Fiedler value , also known as algebraic connectivity, is the
second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler
value among all planar graphs with vertices, denoted by
, and we show the bounds . We also provide bounds on the maximum
Fiedler value for the following classes of planar graphs: Bipartite planar
graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar
graphs. Furthermore, we derive almost tight bounds on for two
more classes of graphs, those of bounded genus and -minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and
Its Application
Efficiency of informational transfer in regular and complex networks
We analyze the process of informational exchange through complex networks by
measuring network efficiencies. Aiming to study non-clustered systems, we
propose a modification of this measure on the local level. We apply this method
to an extension of the class of small-worlds that includes {\it declustered}
networks, and show that they are locally quite efficient, although their
clustering coefficient is practically zero. Unweighted systems with small-world
and scale-free topologies are shown to be both globally and locally efficient.
Our method is also applied to characterize weighted networks. In particular we
examine the properties of underground transportation systems of Madrid and
Barcelona and reinterpret the results obtained for the Boston subway network.Comment: 10 pages and 9 figure
The architecture of the protein domain universe
Understanding the design of the universe of protein structures may provide
insights into protein evolution. We study the architecture of the protein
domain universe, which has been found to poses peculiar scale-free properties
(Dokholyan et al., Proc. Natl. Acad. Sci. USA 99: 14132-14136 (2002)). We
examine the origin of these scale-free properties of the graph of protein
domain structures (PDUG) and determine that that the PDUG is not modular, i.e.
it does not consist of modules with uniform properties. Instead, we find the
PDUG to be self-similar at all scales. We further characterize the PDUG
architecture by studying the properties of the hub nodes that are responsible
for the scale-free connectivity of the PDUG. We introduce a measure of the
betweenness centrality of protein domains in the PDUG and find a power-law
distribution of the betweenness centrality values. The scale-free distribution
of hubs in the protein universe suggests that a set of specific statistical
mechanics models, such as the self-organized criticality model, can potentially
identify the principal driving forces of molecular evolution. We also find a
gatekeeper protein domain, removal of which partitions the largest cluster into
two large sub-clusters. We suggest that the loss of such gatekeeper protein
domains in the course of evolution is responsible for the creation of new fold
families.Comment: 14 pages, 3 figure
Numerical Results for Ground States of Mean-Field Spin Glasses at low Connectivities
An extensive list of results for the ground state properties of spin glasses
on random graphs is presented. These results provide a timely benchmark for
currently developing theoretical techniques based on replica symmetry breaking
that are being tested on mean-field models at low connectivity. Comparison with
existing replica results for such models verifies the strength of those
techniques. Yet, we find that spin glasses on fixed-connectivity graphs (Bethe
lattices) exhibit a richer phenomenology than has been anticipated by theory.
Our data prove to be sufficiently accurate to speculate about some exact
results.Comment: 4 pages, RevTex4, 5 ps-figures included, related papers available at
http://www.physics.emory.edu/faculty/boettcher
Multi-hop Byzantine reliable broadcast with honest dealer made practical
We revisit Byzantine tolerant reliable broadcast with honest dealer algorithms in multi-hop networks. To tolerate Byzantine faulty nodes arbitrarily spread over the network, previous solutions require a factorial number of messages to be sent over the network if the messages are not authenticated (e.g., digital signatures are not available). We propose modifications that preserve the safety and liveness properties of the original unauthenticated protocols, while highly decreasing their observed message complexity when simulated on several classes of graph topologies, potentially opening to their employment
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