1,424 research outputs found

    The minimum rank of universal adjacency matrices

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    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 GG the minimum universal rank of GG is the minimum rank over all matrices of the form U(α,β,γ,δ)=αA+βI+γJ+δD U(\alpha, \beta, \gamma, \delta) = \alpha A + \beta I + \gamma J + \delta D where AA is the adjacency matrix of GG, JJ is the all ones matrix and DD is the matrix with the degrees of the vertices in the main diagonal, and α0,β,γ,δ\alpha\neq 0, \beta, \gamma, \delta 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 AA. 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

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    Given a discrete random walk on a finite graph GG, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step tt.%These sets induce subgraphs of the underlying graph. Let Γ(t)\Gamma(t) be the subgraph of GG induced by the vacant set of the walk at step tt. Similarly, let Γ^(t)\widehat \Gamma(t) be the subgraph of GG induced by the edges of the vacant net. For random rr-regular graphs GrG_r, it was previously established that for a simple random walk, the graph Γ(t)\Gamma(t) of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs Gn,pG_{n,p}. Thus, for r3r \ge 3 there is an explicit value t=t(r)t^*=t^*(r) of the walk, such that for t(1ϵ)tt\leq (1-\epsilon)t^*, Γ(t)\Gamma(t) has a unique giant component, plus components of size O(logn)O(\log n), whereas for t(1+ϵ)tt\geq (1+\epsilon)t^* all the components of Γ(t)\Gamma(t) are of size O(logn)O(\log n). We establish the threshold value t^\widehat t for a phase transition in the graph Γ^(t)\widehat \Gamma(t) of the vacant net of a simple random walk on a random rr-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 rr even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random rr-regular graphs. The main findings are the following: As rr increases the threshold for the vacant set converges to nlogrn \log r in all three walks. For the vacant net, the threshold converges to rn/2  lognrn/2 \; \log n for both the simple random walk and non-backtracking random walk. When r4r\ge 4 is even, the threshold for the vacant net of the unvisited edge process converges to rn/2rn/2, which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk

    Subgraphs in preferential attachment models

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    We consider subgraph counts in general preferential attachment models with power-law degree exponent τ>2\tau>2. For all subgraphs HH, 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

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    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

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    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

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    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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