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Mean Eigenvalue Counting Function Bound for Laplacians on Random NetworksSamavat, Reza 22 January 2015 (has links) (PDF)
Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.
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Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energyHelmberg, Christoph, Trevisan, Vilmar 11 June 2015 (has links) (PDF)
The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.
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On Graph Embeddings and a new Minor Monotone Graph Parameter associated with the Algebraic Connectivity of a GraphWappler, Markus 07 June 2013 (has links) (PDF)
We consider the problem of maximizing the second smallest eigenvalue of the weighted Laplacian of a (simple) graph over all nonnegative edge weightings with bounded total weight.
We generalize this problem by introducing node significances and edge lengths.
We give a formulation of this generalized problem as a semidefinite program.
The dual program can be equivalently written as embedding problem. This is fifinding an embedding of the n nodes of the graph in n-space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by the respective edge length, and the embedded nodes are spread as much as possible. (The sum of the squared norms is maximized.)
We proof the following necessary condition for optimal embeddings. For any separator of the graph at least one of the components fulfills the following property: Each straight-line segment between the origin and an embedded node of the component intersects the convex hull of the embedded nodes of the separator.
There exists always an optimal embedding of the graph whose dimension is bounded by the tree-width of the graph plus one.
We defifine the rotational dimension of a graph. This is the minimal dimension k such that for all choices of the node significances and edge lengths an optimal embedding of the graph can be found in k-space.
The rotational dimension of a graph is a minor monotone graph parameter.
We characterize the graphs with rotational dimension up to two.
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Mean Eigenvalue Counting Function Bound for Laplacians on Random NetworksSamavat, Reza 15 December 2014 (has links)
Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.
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Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energyHelmberg, Christoph, Trevisan, Vilmar 11 June 2015 (has links)
The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.
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On Pairwise Graph ConnectivityHofmann, Tobias 08 August 2023 (has links)
A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality.
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Optimizing Extremal Eigenvalues of Weighted Graph Laplacians and Associated Graph RealizationsReiß, Susanna 09 August 2012 (has links) (PDF)
This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach.
By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem.
Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established.
Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed.
Taking the symmetry of the graph into account, a particular optimal edge weighting exists.
Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.
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Optimizing Extremal Eigenvalues of Weighted Graph Laplacians and Associated Graph RealizationsReiß, Susanna 17 July 2012 (has links)
This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach.
By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem.
Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established.
Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed.
Taking the symmetry of the graph into account, a particular optimal edge weighting exists.
Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.
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On Graph Embeddings and a new Minor Monotone Graph Parameter associated with the Algebraic Connectivity of a GraphWappler, Markus 30 May 2013 (has links)
We consider the problem of maximizing the second smallest eigenvalue of the weighted Laplacian of a (simple) graph over all nonnegative edge weightings with bounded total weight.
We generalize this problem by introducing node significances and edge lengths.
We give a formulation of this generalized problem as a semidefinite program.
The dual program can be equivalently written as embedding problem. This is fifinding an embedding of the n nodes of the graph in n-space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by the respective edge length, and the embedded nodes are spread as much as possible. (The sum of the squared norms is maximized.)
We proof the following necessary condition for optimal embeddings. For any separator of the graph at least one of the components fulfills the following property: Each straight-line segment between the origin and an embedded node of the component intersects the convex hull of the embedded nodes of the separator.
There exists always an optimal embedding of the graph whose dimension is bounded by the tree-width of the graph plus one.
We defifine the rotational dimension of a graph. This is the minimal dimension k such that for all choices of the node significances and edge lengths an optimal embedding of the graph can be found in k-space.
The rotational dimension of a graph is a minor monotone graph parameter.
We characterize the graphs with rotational dimension up to two.:1 Introduction
1.1 Notations and Preliminaries
1.2 The Algebraic Connectivity
1.3 Two applications
1.4 Outline
2 The Embedding Problem
2.1 Semidefinite formulation
2.2 The dual as geometric embedding problem
2.3 Physical interpretation and examples
2.4 Formulation without fifixed barycenter
3 Geometrical Operations
3.1 Congruent transformations
3.2 Folding a flat halfspace
3.3 Folding and Collapsing
4 Structural properties of optimal embeddings
4.1 Separator-Shadow
4.2 Separators containing the origin
4.3 The tree-width bound
4.4 Application to trees
5 The Rotational Dimension of a graph
5.1 Defifinition and basic properties
5.2 Characterization of graphs with small rotational dimension
5.3 The Colin de Verdi ere graph parameter
List of Figures
Bibliography
Theses
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