151,175 research outputs found
Upward-closed hereditary families in the dominance order
The majorization relation orders the degree sequences of simple graphs into
posets called dominance orders. As shown by Hammer et al. and Merris, the
degree sequences of threshold and split graphs form upward-closed sets within
the dominance orders they belong to, i.e., any degree sequence majorizing a
split or threshold sequence must itself be split or threshold, respectively.
Motivated by the fact that threshold graphs and split graphs have
characterizations in terms of forbidden induced subgraphs, we define a class
of graphs to be dominance monotone if whenever no realization of
contains an element as an induced subgraph, and majorizes
, then no realization of induces an element of . We present
conditions necessary for a set of graphs to be dominance monotone, and we
identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
On fractional realizations of graph degree sequences
We introduce fractional realizations of a graph degree sequence and a closely
associated convex polytope. Simple graph realizations correspond to a subset of
the vertices of this polytope. We describe properties of the polytope vertices
and characterize degree sequences for which each polytope vertex corresponds to
a simple graph realization. These include the degree sequences of pseudo-split
graphs, and we characterize their realizations both in terms of forbidden
subgraphs and graph structure.Comment: 18 pages, 4 figure
Approximate entropy of network parameters
We study the notion of approximate entropy within the framework of network
theory. Approximate entropy is an uncertainty measure originally proposed in
the context of dynamical systems and time series. We firstly define a purely
structural entropy obtained by computing the approximate entropy of the so
called slide sequence. This is a surrogate of the degree sequence and it is
suggested by the frequency partition of a graph. We examine this quantity for
standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results
of Pincus, we show that our entropy measure converges with network size to a
certain binary Shannon entropy. On a second step, with specific attention to
networks generated by dynamical processes, we investigate approximate entropy
of horizontal visibility graphs. Visibility graphs permit to naturally
associate to a network the notion of temporal correlations, therefore providing
the measure a dynamical garment. We show that approximate entropy distinguishes
visibility graphs generated by processes with different complexity. The result
probes to a greater extent these networks for the study of dynamical systems.
Applications to certain biological data arising in cancer genomics are finally
considered in the light of both approaches.Comment: 11 pages, 5 EPS figure
Push is Fast on Sparse Random Graphs
We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every , whp rounds are
sufficient to inform all but an -fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with , the push-pull protocol needs
to inform all but vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with , push-pull is exponentially faster than pull
On 2-switches and isomorphism classes
A 2-switch is an edge addition/deletion operation that changes adjacencies in
the graph while preserving the degree of each vertex. A well known result
states that graphs with the same degree sequence may be changed into each other
via sequences of 2-switches. We show that if a 2-switch changes the isomorphism
class of a graph, then it must take place in one of four configurations. We
also present a sufficient condition for a 2-switch to change the isomorphism
class of a graph. As consequences, we give a new characterization of matrogenic
graphs and determine the largest hereditary graph family whose members are all
the unique realizations (up to isomorphism) of their respective degree
sequences.Comment: 11 pages, 6 figure
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