1,424 research outputs found
The minimum rank of universal adjacency matrices
In this paper we introduce a new parameter for a graph called the {\it
minimum universal rank}. This parameter is similar to the minimum rank of a
graph. For a graph the minimum universal rank of is the minimum rank
over all matrices of the form where is the adjacency matrix of ,
is the all ones matrix and is the matrix with the degrees of the vertices
in the main diagonal, and are scalars.
Bounds for general graphs based on known graph parameters are given, as is a
formula for the minimum universal rank for regular graphs based on the
multiplicity of the eigenvalues of . The exact value of the minimum
universal rank of some families of graphs are determined, including complete
graphs, complete bipartite graph, paths and cycles. Bounds on the minimum
universal rank of a graph obtained by deleting a single vertex are established.
It is shown that the minimum universal rank is not monotone on induced
subgraphs, but bounds based on certain induced subgraphs, including bounds on
the union of two graphs, are given. Finally we characterize all graphs with
minimum universal rank equal to 0 and to 1
Vacant sets and vacant nets: Component structures induced by a random walk
Given a discrete random walk on a finite graph , the vacant set and vacant
net are, respectively, the sets of vertices and edges which remain unvisited by
the walk at a given step .%These sets induce subgraphs of the underlying
graph. Let be the subgraph of induced by the vacant set of the
walk at step . Similarly, let be the subgraph of
induced by the edges of the vacant net. For random -regular graphs , it
was previously established that for a simple random walk, the graph
of the vacant set undergoes a phase transition in the sense of the phase
transition on Erd\H{os}-Renyi graphs . Thus, for there is an
explicit value of the walk, such that for ,
has a unique giant component, plus components of size ,
whereas for all the components of are of
size . We establish the threshold value for a phase
transition in the graph of the vacant net of a simple
random walk on a random -regular graph. We obtain the corresponding
threshold results for the vacant set and vacant net of two modified random
walks. These are a non-backtracking random walk, and, for even, a random
walk which chooses unvisited edges whenever available. This allows a direct
comparison of thresholds between simple and modified walks on random
-regular graphs. The main findings are the following: As increases the
threshold for the vacant set converges to in all three walks. For
the vacant net, the threshold converges to for both the simple
random walk and non-backtracking random walk. When is even, the
threshold for the vacant net of the unvisited edge process converges to ,
which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk
Subgraphs in preferential attachment models
We consider subgraph counts in general preferential attachment models with
power-law degree exponent . For all subgraphs , we find the scaling
of the expected number of subgraphs as a power of the number of vertices. We
prove our results on the expected number of subgraphs by defining an
optimization problem that finds the optimal subgraph structure in terms of the
indices of the vertices that together span it and by using the representation
of the preferential attachment model as a P\'olya urn model
Characterization of Large Scale Functional Brain Networks During Ketamine-Medetomidine Anesthetic Induction
Several experiments evidence that specialized brain regions functionally
interact and reveal that the brain processes and integrates information in a
specific and structured manner. Networks can be used to model brain functional
activities constituting a way to characterize and quantify this structured form
of organization. Reports state that different physiological states or even
diseases that affect the central nervous system may be associated to
alterations on those networks, that might reflect in graphs of different
architectures. However, the relation of their structure to different states or
conditions of the organism is not well comprehended. Thus, experiments that
involve the estimation of functional neural networks of subjects exposed to
different controlled conditions are of great relevance. Within this context,
this research has sought to model large scale functional brain networks during
an anesthetic induction process. The experiment was based on intra-cranial
recordings of neural activities of an old world macaque of the species Macaca
fuscata. Neural activity was recorded during a Ketamine-Medetomidine anesthetic
induction process. Networks were serially estimated in time intervals of five
seconds. Changes were observed in various networks properties within about one
and a half minutes after the administration of the anesthetics. These changes
reveal the occurrence of a transition on the networks architecture. During
general anesthesia a reduction in the functional connectivity and network
integration capabilities were verified in both local and global levels. It was
also observed that the brain shifted to a highly specific and dynamic state.
The results bring empirical evidence and report the relation of the induced
state of anesthesia to properties of functional networks, thus, they contribute
for the elucidation of some new aspects of neural correlates of consciousness.Comment: 28 pages , 9 figures, 7 tables; - English errors were corrected;
Figures 1,3,4,5,6,8 and 9 were replaced by (exact the same)figures of higher
resolution; Three(3) references were added on the introduction sectio
Constraint satisfaction problems for reducts of homogeneous graphs
For n >= 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Gamma is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete
Constraint Satisfaction Problems for Reducts of Homogeneous Graphs
For n >= 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Gamma is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete
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