89 research outputs found
Family of Circulant Graphs and Its Expander Properties
In this thesis, we apply spectral graph theory to show the non-existence of an
expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger\u27s inequalities and determine when the equalities hold. In order to apply Cheeger\u27s inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family
Random unitary matrices associated to a graph
We analyze composed quantum systems consisting of subsystems, each
described by states in the -dimensional Hilbert space. Interaction between
subsystems can be represented by a graph, with vertices corresponding to
individual subsystems and edges denoting a generic interaction, modeled by
random unitary matrices of order . The global evolution operator is
represented by a unitary matrix of size . We investigate statistical
properties of such matrices and show that they display spectral properties
characteristic to Haar random unitary matrices provided the corresponding graph
is connected. Thus basing on random unitary matrices of a small size one
can construct a fair approximation of large random unitary matrices of size
. Graph--structured random unitary matrices investigated here allow one
to define the corresponding structured ensembles of random pure states.Comment: 13 pages, 10 figures, 1 tabl
Multilevel Artificial Neural Network Training for Spatially Correlated Learning
Multigrid modeling algorithms are a technique used to accelerate relaxation
models running on a hierarchy of similar graphlike structures. We introduce and
demonstrate a new method for training neural networks which uses multilevel
methods. Using an objective function derived from a graph-distance metric, we
perform orthogonally-constrained optimization to find optimal prolongation and
restriction maps between graphs. We compare and contrast several methods for
performing this numerical optimization, and additionally present some new
theoretical results on upper bounds of this type of objective function. Once
calculated, these optimal maps between graphs form the core of Multiscale
Artificial Neural Network (MsANN) training, a new procedure we present which
simultaneously trains a hierarchy of neural network models of varying spatial
resolution. Parameter information is passed between members of this hierarchy
according to standard coarsening and refinement schedules from the multiscale
modelling literature. In our machine learning experiments, these models are
able to learn faster than default training, achieving a comparable level of
error in an order of magnitude fewer training examples.Comment: Manuscript (24 pages) and Supplementary Material (4 pages). Updated
January 2019 to reflect new formulation of MsANN structure and new training
procedur
Connectivity of generating graphs of nilpotent groups
Let be -generated group. The generating graph of is the
graph whose vertices are the elements of and where two vertices and
are adjacent if . This graph encodes the combinatorial
structure of the distribution of generating pairs across . In this paper we
study several natural graph theoretic properties related to the connectedness
of in the case where is a finite nilpotent group. For example,
we prove that if is nilpotent, then the graph obtained from by
removing its isolated vertices is maximally connected and, if ,
also Hamiltonian. We pose several questions.Comment: 11 pages; to appear in Algebraic Combinatoric
Spectral behavior of some graph and digraph compositions
Let G be a graph of order n the vertices of which are labeled from 1 to n and let , · · · , be n graphs. The graph composition G[, · · · ,] is the graph obtained by replacing the vertex i of G by the graph Gi and there is an edge between u ∈ and v ∈ if and only if there is an edge between i and j in G. We first consider graph composition G[, · · · ,] where G is regular and is a complete graph and we establish
some links between the spectral characterisation of G and the spectral characterisation of G[, · · · ,]. We then prove that two non isomorphic graphs G[, · · ·] where are complete
graphs and G is a strict threshold graph or a star are not Laplacian-cospectral, giving rise to a spectral characterization
of these graphs. We also consider directed graphs, especially the vertex-critical tournaments without non-trivial acyclic interval which are tournaments of the shape t[, · · · ,], where t
is a tournament and is a circulant tournament. We give
conditions to characterise these graphs by their spectrum.Peer Reviewe
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