9 research outputs found
On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
Courant theorem provides an upper bound for the number of nodal domains of
eigenfunctions of a wide class of Laplacian-type operators. In particular, it
holds for generic eigenfunctions of quantum graph. The theorem stipulates that,
after ordering the eigenvalues as a non decreasing sequence, the number of
nodal domains of the -th eigenfunction satisfies . Here,
we provide a new interpretation for the Courant nodal deficiency in the case of quantum graphs. It equals the Morse index --- at a
critical point --- of an energy functional on a suitably defined space of graph
partitions. Thus, the nodal deficiency assumes a previously unknown and
profound meaning --- it is the number of unstable directions in the vicinity of
the critical point corresponding to the -th eigenfunction. To demonstrate
this connection, the space of graph partitions and the energy functional are
defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure
Critical partitions and nodal deficiency of billiard eigenfunctions
The paper addresses the the number of nodal domains for eigenfunctions of
Schr\"{o}dinger operators with Dirichlet boundary conditions in bounded
domains. In dimension one, the th eigenfunction has nodal domains. The
Courant Theorem claims that in any dimension, the number of nodal domains of
the th eigenfunction cannot exceed . However, in dimensions higher than 1
the equality can hold for only finitely many eigenfunctions. Thus, a "nodal
deficiency" arises. Examples are known of eigenfunctions with arbitrarily large
index that have just two nodal domains.
It was suggested in the recent years to look at the partitions of the domain,
rather than eigenfunctions. It was shown in a recent paper by Helffer,
Hoffmann-Ostenhof and Terracini that (under some natural conditions) bipartite
partitions minimizing the maximum of the ground-state energies in sub-domains
of the partition, correspond to the "Courant sharp" eigenfunctions, i.e. to
those with zero nodal deficiency.
In this paper, the authors show, under some genericity conditions, among the
bipartite equipartitions, the nodal ones correspond exactly to the critical
points of an analogous functional, with the nodal deficiency being equal to the
Morse index at this point. This explains, in particular, why all the minimal
partitions must be Courant sharp.Comment: In the 2nd version minor modifications were implemente
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815-838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic-rather than numerical-results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45-72, 2005), Helffer et al. (Ann Inst Henri Poincare Anal Non Lineaire 26:101-138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions
Zeros of Eigenfunctions of the Schrodinger Operator on Graphs and Their Relation to the Spectrum of the Magnetic Schrodinger Operator
In this dissertation, we analyze characteristics of eigenfunctions of the Schrödinger operator on graphs. In particular, we are interested in the zeros of the eigenfunctions and their relation to the spectrum of the magnetic Schrödinger operator.
We begin by studying the nodal count on finite quantum graphs, analyzing both the number and location of the zeros of eigenfunctions. This question was completely solved by Sturm in one dimension. In higher dimensions (including domains and graphs), we only know bounds for the nodal count. We discover more information about the nodal count on quantum graphs while analyzing eigenvalues of the magnetic Schrödinger operator. In particular, we show a relation between the stability of eigenvalues of the magnetic Schrödinger operator with respect to magnetic flux and the number of zeros of the corresponding eigenfunctions. We also study the location of the zeros of eigenfunctions while analyzing partitions. Specifically, we show that the critical points of the energy functional are the nodal partitions corresponding to zeros of an eigenfunction and that the stability of these critical points is related to the nodal count.
Then using Floquet-Bloch theory, we study the spectrum of the Schrödinger operator on infinite periodic graphs by analyzing the eigenvalues of the magnetic Schrödinger operator on a fundamental domain. Here we consider both discrete and quantum graphs. We find a characterization of critical points of the dispersion relation that occur inside the Brillouin zone under certain conditions on the graph. In particular, we show that if the fundamental domain is a tree, then the eigenfunction corresponding to an interior critical point must be zero on a vertex.
Finally, we use the results from infinite periodic graphs to study the magnetic Schrödinger operator on a finite quantum d-mandarin graph. We find that extremal points of the dispersion surface occur inside the Brillouin zone where two surfaces touch and the corresponding eigenfunction is zero on a vertex
Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs
In this thesis we investigate three different aspects of graph theory.
Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs.
Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber.
Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses.
Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a âdualâ version of the reconstruction conjecture
HPC techniques for large scale data analysis
In the present work we apply High-Performance Computing techniques to two Big Data problems. The frst one deals with the analysis of large graphs by using a parallel distributed architecture, whereas the second one consists in the design and implementation of a scalable solution for fast indexing and searching of large datasets of heterogeneous documents