Spectral Graph Theory

DOI: 10.1103/PhysRevE.83.046114

A spectral algorithm for community detection is presented. The algorithm consists of three stages: (1) matrix factorization of two matrix forms, square signless Laplacian for unipartite graphs and rectangular adjacency matrix for bipartite graphs, using singular value decompostion (SVD); (2) dimensionality reduction using an optimal linear approximation; and (3) clustering vertices using dot products in reduced dimensional space. The algorithm reveals communities in graphs without placing any restriction on the input network type or the output community type. It is applicable on unipartite or bipartite unweighted or weighted networks. It places no requirement on strict community membership and automatically reveals the number of clusters, their respective sizes and overlaps, and hierarchical modular organization. By representing vertices as vectors in real space, expressed as linear combinations of the orthogonal bases described by SVD, orthogonality becomes the metric for classifying vertices into communities. Results on several test and real world networks are presented, including cases where a mix of disjointed, overlapping, or hierarchical communities are known to exist in the network.

The spectral lines given off by Hydrogen are well known and is simply described by the Rydberg formula. However, this only works on the hydrogen atom. If we try to describe the spectra with the Rydberg formula for helium and lithium, it fails – or does it? This a paper explores the idea that the spectra of heavier elements like helium and lithium can actually be described by just providing scaling factors to the Rydberg formula to explain the spectra given off by multi-electron atoms. This shows that the spectra is not a complex multi-body problem and that there is a very simple stair case pattern that the spectra follows.

Let G be a bipartite graph of order n with m edges. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for E(G) in terms of n, m, and det A. Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.

Résumé Nous nous intéressons dans cet article au problème passionnant qu'est la reconnaissance de visages dans des cas non contraints, c'est-à-dire dans des situations où l'éclairage, la pose, la taille du visage dans l'image ne sont pas contrôlés. Nous proposons ici deux contributions originales: une méthode d'apprentissage de distance par analyse en composantes logistiques discriminantes (LDCA), combinée à une méthode d'apprentissage semi-supervisé au moyen de graphes.

For a simple graph G of order n, size m and with signless Laplacian eigenvalues q 1 , q 2 ,. .. , q n , the signless Laplacian energy QE(G) is defined as QE(G) = n i=1 |q i − d|, where d = 2m n is the average vertex degree of G. We obtain the lower bounds for QE(G), in terms of first Zagreb index M 1 (G), maximum degree d 1 , second maximum degree d 2 , minimum degree d n and second minimum degree d n−1. As a consequence of these bounds, we obtain several bounds for the energy E(L (G)) of the line graph L (G) of graph G in terms of various graph parameters like M 1 (G), ω (the clique number), n, m, etc., which improve some recently known bounds.

Our goal in this paper is to show that many of the tools of signal processing, adapted Fourier and wavelet analysis can be naturally lifted to the setting of digital data clouds, graphs and manifolds. We use diffusion as a smoothing and scaling tool to enable coarse graining and multiscale analysis. Given a diffusion operator T on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing T t . This allows a direct multiscale computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. Classes of operators for which these computations are fast include certain diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. We use ideas related to the Fast Multipole Methods, and to the wavelet analysis of Calderón-Zygmund and pseudodifferential operators, to numerically enforce the emergence of a natural hierarchical coarse graining of a manifold, graph or data set. For example for a body of text documents the construction leads to a directory structure at different levels of generalization. The dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory: we construct, with efficient and stable algorithms, bases of orthonormal scaling functions and wavelets associated to this multiresolution analysis, together with the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. While most of our discussion deals with symmetric operators and relates to localization to spectral bands, the symmetry of the operators and their spectral theory need not be considered, as the main assumption is reduction of the numerical ranks as we take powers of the operator.

In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be D Q S if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a D Q S graph G, we show that G ∪ r K 1 ∪ s K 2 is D Q S under certain conditions, where r , s are natural numbers and K 1 andK 2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some D Q S graphs with independent edges and isolated vertices are obtained.

This paper deals with graphs that are known as multicone graphs. A multicone graph is a graph obtained from the join of a clique and a regular graph. Let w, l, m be natural numbers and k is a natural number. It is proved that any connected graph cospectral with multicone graph Kw mECP k l is determined by its adjacency spectra as well as its Laplacian spectra, where ECP k l = K 3 k , 3 k , ..., 3 k l times. Also, we show that complements of some of these mul-ticone graphs are determined by their adjacency spectra. Moreover, we prove that any connected graph cospectral with these multicone graphs must be perfect. Finally , we pose two problems for further researches. Mathematics Subject Classification (2010): 05C50.

Consider a mobile robot exploring an initially unknown school building and assume that it has already discovered some classrooms, offices, and bathrooms. What can the robot infer about the presence and the locations of other classrooms and offices in the school building? This thesis makes a step toward providing an answer to the above question by proposing a system based on a generative model that is able to represent the topological structures and the semantic labeling schemas of buildings and to predict the structure and the schema for the unexplored portions of these environments. Buildings are  represented as undirected graphs, whose nodes are labeled with names of rooms and edges are physical connections between them. Given an initial knowledge base of graphs, the proposed approach, relying on a spectral analysis of these graphs, segments each graph for finding significant subgraphs, which are then clustered according to their similarity. A graph representing a new building or an unvisited part of a building is eventually generated by sampling subgraphs from clusters and connecting them.

In the subject of Chemistry, the total π-electron energy of a chemical-molecular graph has a long-known application . This quantity has been studied for decades, in particular by mathematicians who also specialise in Chemistry. In the last two decades, this quantity has become more apparent in the mathematical literature . I found a paper which was published in 2006 by: “Ivan Gutman” and “Bo Zhou”, titled as: “The Laplacian Energy of a Graph” (see: [2]) . After studying this paper, I noticed a great deal of analogy between the earlier defined energy and the laplacian energy, as well as a number of significant differences.
  The main goal of this dissertation is to represent a modified version of the proofs given in [2] for obtaining useful upper and lower-bounds for the laplacian energy. We will be showing the important role that the average vertex-degree plays in the proofs and formalisations rather than the use of the size of the graph (the number of edges).        After studying the preliminaries and being introduced to the energy of a graph, the laplacian energy will be defined and some of the important properties of this quantity will be investigated.

A lot of attempts have been made in recent years to generate geometrically correct floor plans, spatial configurations and urban layouts in connection with functional relations and defined spatial properties (Elezkurtaj and Franck, 1999) (Duarte, 2001) (Donath, König and Petzold, 2012) (Nourian, Rezvani and Sariyildiz, 2013). Different heuristics (force based drawing...) / optimisations methods (metaheuristic solvers such as genetic algorithms, simulated annealing...) have been applied to search for the »best« trade of between a set of constraint/properties. Most of these techniques are based on an iterative and time-consuming process allowing to find a good solution for one, two, maybe three different constraints/properties. But architecture is related to a multitude of different constraints/properties, which strongly depend on the given task and its context.
In image processing spectral graph theory has shown to solve different tasks such as graph drawing, image matching and image segmentation. A direct translation to architecture seems obvious as an image similar to a plan represents a particular spatial embedding of elements (pixels, rooms...) in two-dimensional space. As these problems are described through graphs, which represent the relation of elements rather than a particular spatial embedding, these approaches are applicable not only to a two-dimensional space but also in higher dimensions.
With such tools at hand, a prototype of a spatial configuration defined by a graph (functional or through properties such as e.g. integration, choice...) can be applied to an existing configuration (e.g. the refurbishment of existing buildings), to a spatial configuration which is generated by other properties than the prototype (e.g. structural, solar gain...) or to the most generic: A grid, resulting in a possible spatial embedding of the prototype. This allows to unlink functional / configurational constraint with other properties (an existing configuration, other configurational constraints/properties such as structural, solar...). Through unlinking, each constraint can be looked at on its own, developed on its own to a wanted solution and than relinked again. Further the process of matching two graphs, based on spectral graph theory, is not based on an iterative process but on one with a fixed
computational time. Here the bottleneck is the calculation of the eigenvectors which can be preformed at least in O(n3).

is determined by its signless Laplacian spectra under certain conditions, where r and K 2 denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

Identification and authentication of multimedia content has become one of the most important aspects of multimedia security. In this paper, we present a hashing technique for 3D models using spectral graph theory and entropic spanning trees. The main idea is to partition a 3D triangle mesh into an ensemble of sub-meshes, then apply eigen-decomposition to the Laplace-Beltrami matrix of each sub-mesh, followed by computing the hash value of each sub-mesh. This hash value is defined in terms of spectral coefficients and Tsallis entropy estimate. The experimental results on a variety of 3D models demonstrate the effectiveness of the proposed technique in terms of robustness against the most common attacks including Gaussian noise, mesh smoothing, mesh compression, scaling, rotation as well as combinations of these attacks.

Knödel graphs have, of late, come to be used as strong competitors for  hypercubes in the realms of broadcasting and gossiping in interconnection  networks. For an even positive integer 'n' and 1 ≤ Δ ≤  ⌊log₂ n⌋, the general Knödel graph W_{Δ, n} is the Δ-regular bipartite graph with bipartition sets X = {x₀, x₁, …, x_{n/2 - 1}} and Y = {y₀, y₁, …, y_{n/2  - 1}} such that xⱼ is adjacent to yⱼ, y_{j + 2^1 - 1}, y_{j+2^2-1}, …, y_{j+2^{Δ-1}-1}, with suffixes being taken modulo n/2. The edge  x_j y_{j+2^i-1} at xⱼ and the edge y_j x_{j-(2^i-1)} at $y_j$ are called  edges of  dimension i at the stars centered at xⱼ and yⱼ respectively. In this paper, we concentrate on the Knödel graph Wₖ = W_{k, 2^k} with k ≥ 4. We show that for k ≥ 4, any automorphism of Wₖ fixes the set of 0-dimensional edges of Wₖ.  We determine the automorphism group Aut(Wₖ) of Wₖ and show  that it is isomorphic to the dihedral group D_{2^{k-1}}. In addition, we  determine the spectrum of Wₖ and prove that it is never integral. As a by-product of our results, we obtain three new proofs showing that, for k ≥ 4, Wₖ is not isomorphic to the hypercube Hₖ of dimension k, and a new
proof for the result that Wₖ is not edge-transitive.

This is week 8 of the lecture notes series in Computational methods for Economists.
This week we looked at sub-graph densities and clustering measures that allow us to distinguish between different types of networks and find clues concerning how networks originated and evolved. We cover:
- degree- and subgraph-density correlations
- Watts-Strogatz clustering;
- Newman clustering index;
After that we turn towards a motivation that economic agents in networks will have incentives to amend the links when solving optimal allocation problems on a network (for example) and use this to make a start with a exploration of network "re-wiring" and preferential attachment models for network evolution.

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.

We define and study the link prediction problem in bipartite networks, specializing general link prediction algorithms to the bipartite case. In a graph, a link prediction function of two vertices denotes the similarity or proximity of the vertices. Common link prediction functions for general graphs are defined using paths of length two between two nodes. Since in a bipartite graph adjacency vertices can only be connected by paths of odd lengths, these functions do not apply to bipartite graphs. Instead, a certain class of graph kernels (spectral transformation kernels) can be generalized to bipartite graphs when the positivesemidefinite kernel constraint is relaxed. This generalization is realized by the odd component of the underlying spectral transformation. This construction leads to several new link prediction pseudokernels such as the matrix hyperbolic sine, which we examine for rating graphs, authorship graphs, folksonomies, document-feature networks and other types of bipartite networks.

Let G be a graph of order n with m edges and clique number ω. Let μ 1 ≥ μ 2 ≥. .. ≥ μ n = 0 be the Laplacian eigenvalues of G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that μ σ ≥ 2m n. In this paper we study the relation between σ and ω. In particular, we provide the answer to Problem 2.3 raised in Pirzada and Ganie (2015) [15]. Moreover, we characterize all connected threshold graphs with σ < ω − 1, σ = ω − 1 and σ > ω − 1. We obtain Nordhaus– Gaddum-type results for σ. Some relations between σ with other graph invariants are obtained.

We introduce and study the spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices: In large networks, changes over time result in a change of a graph's spectrum, leaving the eigenvectors unchanged. We validate this hypothesis for several large social, collaboration, authorship, rating, citation, communication and tagging networks, covering unipartite, bipartite, signed and unsigned graphs. Following these observations, we introduce a link prediction algorithm based on the extrapolation of a network's spectral evolution. This new link prediction method generalizes several common graph kernels that can be expressed as spectral transformations. In contrast to these graph kernels, the spectral extrapolation algorithm does not make assumptions about specific growth patterns beyond the spectral evolution model. We thus show that it performs particularly well for networks with irregular, but spectral, growth patterns.

Messages routing over a network is one of the most fundamental concept in communication which requires simultaneous transmission of messages from a source to a destination. In terms of Real-Time Routing, it refers to the addition of a timing constraint in which messages should be received within a specified time delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of routing real-time messages on networks. We study whether it is possible to have an optimal on-line algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message routing's algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual examples. Next, we explore the Unidirectional Ring topology in finding the transmission's " makespan " .Lastly, we propose the same network modelling through the technique of Kinesthetic Learning Activity (KLA). We will analyse the data collected and present the results in a case study to evaluate the effectiveness of the KLA approach compared to the traditional teaching method.

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright a b s t r a c t Video surveillance is becoming the technology of choice for monitoring crowded areas for security threats. While video provides ample information for human inspectors, there is a great need for robust automated techniques that can efficiently detect anomalous behavior in streaming video from single or multiple cameras. In this work we synergistically combine two state-of-the-art methodologies. The first is the ability to track and label single person trajectories in a crowded area using multiple video cameras , and the second is a new class of novelty detection algorithms based on spectral analysis of graphs. By representing the trajectories as sequences of transitions between nodes in a graph, shared individual trajectories capture only a small subspace of the possible trajectories on the graph. This subspace is characterized by large connected components of the graph, which are spanned by the eigenvectors with the low eigenvalues of the graph Laplacian matrix. Using this technique, we develop robust invariant distance measures for detecting anomalous trajectories, and demonstrate their application on real video data.

In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets. We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient representations of complex geometric structures. The associated family of diffusion distances, obtained by iterating the Markov matrix, defines multiscale geometries that prove to be useful in the context of data parametrization and dimensionality reduction. The proposed framework relates the spectral properties of Markov processes to their geometric counterparts and it unifies ideas arising in a variety of contexts such as machine learning, spectral graph theory and eigenmap methods.

In this paper, we investigate the properties of the cell-graphs by using the spectral graph theory. The spectral analysis is performed on (i) the adjacency matrix of a cell-graph, and (ii) the normalized Laplacian of the cell-graph. We show that the spectra of the cell-graphs of cancerous tissues are unique and the features extracted from these spectra distinguish the cancerous (malignant glioma) tissues from the healthy and benign reactive/inflammatory processes. Our experiments on 646 brain biopsy samples of 60 different patients demonstrate that by using spectral features defined on the normalized Laplacian of the graph, we achieve 100% accuracy in the classification of cancerous and healthy tissues. In the classification of cancerous and benign tissues, our experiments yield 92% and 89% accuracy on the testing set for the cancerous and benign tissues. We also analyze graph spectra to identify the distinctive spectral features of the cancerous tissues to conclude that (i) the features representing the cellular density are the most distinctive features to distinguish the cancerous and healthy tissues, (ii) and the number of the eigenvalues in the normalized Laplacian spectrum that have a value of 0 is the most distinctive feature to distinguish the cancerous and benign tissues.

This paper proposes a novel non-parametric technique for clustering networks based on their structure. Many topological measures have been introduced in the literature to characterize topological properties of networks. These measures provide meaningful information about the structural properties of a network, but many networks share similar values of a given measure . Furthermore, strong correlation between these measures occur on real-world graphs [2], so that using them to distinguish arbitrary graphs is difficult in practice .

Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent K-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of
embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.

Power grids are generally regarded as very reliable systems, nevertheless outages and electricity shortfalls are common events. Severe accidents have the potential to produce significant social and economic consequences, hence it is important to reduce their likelihood by assuring safe operations and robust topologies. Grid safety relies on accurate vulnerability measures, control schemes and good quality information. For instance during the network operations , contingency analysis is used to constrain the network to secure operative states with respect to predefined failures (e.g. list of single component failures). In order to better understand the power network weakness and strengths a variety of robustness metrics have been introduced in literature. However, many of the proposed vulnerability metrics lack in considering uncertainties which can affect results and goodness of analysis. In this work power network vulnerability to failure events is analysed and outage events have been ranked using different metrics under different assumptions (e.g. topology-based, flow-based). Single line contingencies (N-1) have been accounted for and sources of uncertainty such as stochastic behaviour of the power demand and lack of precise knowledge on the component parameters have been considered in all the phases of the calculation. The assumption underpinning the methodolo-gies and the results are discussed and metrics compared also after uncertainty quantification performed. A modified version of the IEEE 118 bus power network has been selected as representative case study. Different robustness indexes are compared by taking into account sources of uncertainties in the calculations.

a security scheme of the power systems should protect the integrity of the electric networks and carry out fast operations on the entire power system to prevent a possible blackout. Blackouts started as local failures led to electrical disturbances, and the complete collapse of the power systems. This work proposes a security scheme based in two stages. In the first stage, the controlled separation of the power system allows to isolate the failure area. In the second stage, under frequency load shedding control actions should be applied to maintain a balance between consumption and generation power. This work presents some schemes in the IEEE-39 test system.

Ears are complicated shapes and contain a lot of folds. It is difficult to correctly deform an ear template to achieve the same shape as a scan while avoiding to reconstructing the noise from the scan and be robust to bad geometry found in the scan. We leverage the smoothness of the spectral space to help in the alignment of the semantic features of the ears. Edges detected in image space are used to identify relevant features from the ear that we align in the spectral representation by iteratively deforming the template ear. We then apply a novel reconstruction that preserves the deformation from the spectral space while reintroducing the original details. A final deformation based on constraints considering surface position and orientation deforms the template ear to match the shape of the scan. We tested our approach on many ear scans and observed that the resulting template shape provides a good compromise between complying to the shape of the scan and avoiding to reconstruct the noise found in the scan. Furthermore, our approach was robust to scan meshes exhibiting typical bad geometry such as cracks and handles.

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ[subscript 1],…,μ[subscript n] be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound χ ≥ 1 + max[subscript m]{∑[m over i=1]μ[subscript i]/ − ∑[m over i=1]μ[subscript n−i+1]} for m = 1,…,n − 1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case m = 1. We provide several examples for which the new bound exceeds the Hoffman lower bound. Second, we conjecture the lower bound χ ≥ 1 + s[superscript +/s[superscript −], where s[superscript +] and s[superscript −] are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the bound χ ≥ s[superscript +]/s[superscript −]. We show that the conjectured lower bound is true for several families of graphs. We also performed various searches for a counter-example, but none was foun...

LetGbeaconnectedthresholdgraphofordernwithmedgesandtraceT.InthispaperwegivealowerboundonLaplacianenergyintermsofn,mandTofG.FromthiswedeterminethethresholdgraphswiththefirstfourminimalLaplacianenergies.Moreover,weobtainthethresholdgraphswiththelargestandthesecondlargestLaplacianenergies.

Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices, and let m be a positive integer number. In A. Z. Abdian (2016) it has been shown that multicone graphs Kw ▽ P 17 and Kw ▽ S are determined by both their adjacency and Laplacian spectra, where P 17 and S denote the Paley graph of order 17 and the Schläfli graph, respectively. In this paper, we generalize these results and we prove that multicone graphs Kw ▽ mP 17 and Kw ▽ mS are determined by their adjacency spectra as well as their Laplacian spectra.

A resistance network is a connected graph (G, c). The conductance function c xy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E produces a Hilbert space structure (which we call the energy space H E ) on the space of functions of finite energy.

In this work we deal with the characteristic polynomial of the Laplacian of a graph. We present some general results about the coefficients of this polynomial. We present families of graphs, for which the number of edges m is given by a linear function of the number of vertices n. In some of these graphs we can find certain coefficients of the above-named polynomial as functions just of n.

Diffusion wavelets can be constructed on manifolds, graphs and allow an efficient multiscale representation of powers of a diffusion operator acting on their domain. While diffusion wavelets are expected to perform rather well in general, in many applications it is necessary to have more versatility in choosing a basis in which to analyze, denoise, compress and manipulate a signal. Wavelet packets have proven very successful in many applications, ranging from image denoising, 2 and 3 dimensional compression of data (e.g. images, seismic data, hyperspectral data) and in discrimination tasks as well. Till now these tools for signal processing have been available only in Euclidean settings, and in low dimensions. Building upon the recent construction of diffusion wavelets, we show how to construct diffusion wavelet packets, generalizing the classical construction of wavelet packets, and allowing the same algorithms existing in the Euclidean setting to be lifted to rather general geometric and anisotropic settings, in higher dimension, on manifolds, graphs and even more general spaces. We show fast algorithms exists for computations involving these objects, discuss some applications and examples.

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ℓ 2 (X), and the energy space H E . In particular, we prove that these operators are always essentially self-adjoint on ℓ 2 (X), but may fail to be essentially self-adjoint on H E . In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the H E operators with the use of a new approximation scheme. also . In modern language, this may be phrased as a specific representation of the 1 two operators P and Q in a Hilbert space of quantum states in matrix form. The matrix 2 form in turn is computed with respect to a particular choice of orthonormal basis (onb).

This paper proposes a new robust 3-D object blind watermarking method using constraints in the spectral domain. Mesh watermarking in spectral domain has the property of spreading the information in unpredictable ways, thus increasing the security of the watermark. In the proposed method, firstly, the Laplacian matrix of the graphical object mesh is eigen-decomposed. The coefficients corresponding to the higher spectra are split into sets and each set is used for embedding one bit. A bit of 1 is embedded by introducing an asymmetry in the 3-D distribution of the spectral coefficients from the given set, while the distribution symmetry is enforced in the case when embedding a bit of 0. The Principal Component Analysis (PCA) is used for embedding the constraints in the spectral domain by ensuring a minimal distortion. Comparison results are provided for various attacks.

E. R. van Dam and W. H. Haemers [15] conjectured that almost all graphs are determined by their spectra. Nevertheless, the set of graphs which are known to be determined by their spectra is small. Hence, discovering infinite classes of graphs that are determined by their spectra can be an interesting problem. The aim of this paper is to characterize new classes of multicone graphs that are determined by their spectrum. A multicone graph is defined to be the join of a clique and a regular graph. It is proved that any graph cospectral with multicone graph Kw L(P) is determined by its adjacency spectrum as well as its Laplacian spectrum, where Kw and L(P) denote a complete graph on w vertices and the line graph of the Petersen graph, respectively. Finally, three problems for further researches are proposed.

With ever growing databases containing multimedia data, indexing has become a necessity to avoid a linear search. We propose a novel technique for indexing multimedia databases in which entries can be represented as graph structures. In our method, the topological structure of a graph as well as that of its subgraphs are represented as vectors whose components correspond to the sorted laplacian eigenvalues of the graph or subgraphs. Given the laplacian spectrum of graph G, we draw from recently developed techniques in the field of spectral integral variation to generate the laplacian spectrum of graph G þ e without computing its eigendecomposition, where G þ e is a graph obtained by adding edge e to graph G. This process improves the performance of the system for generating the subgraph signatures for 1.8% and 6.5% in datasets of size 420 and 1400, respectively. By doing a nearest neighbor search around the query spectra, similar but not necessarily isomorphic graphs are retrieved. Given a query graph, a voting schema ranks database graphs into an indexing hypothesis to which a final matching process can be applied. The novelties of the proposed method come from the powerful representation of the graph topology and successfully adopting the concept of spectral integral variation in an indexing algorithm. To examine the fitness of the new indexing framework, we have performed a number of experiments using an extensive set of recognition trials in the domain of 2D and 3D object recognition. The experiments, including a comparison with a competing indexing method using two different graph-based object representations, demonstrate both the robustness and efficacy of the overall approach.

We present an original approach to cluster multi-component datasets, in the frame of two applications: taxonomic classification of asteroids and clustering of chemical species on a Mars hyperspectral image. The method is based on an estimation of the number of clusters and an initialization of clusters centroids in order to partition the data with a popular clustering algorithm: K-means. These

A resistance network is a connected graph (G, c). The conductance function c xy weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form E and the discrete Laplace operator ∆ on a finite network is given by E(u, v) = u, ∆v 2 , where the latter is the usual ℓ 2 inner product. We extend this formula to infinite networks, where a new (boundary) term appears. The Laplace operator is typically unbounded in this context; we construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for ∆ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel also yield a boundary integral representation for harmonic functions of finite energy; this representation is the focus of a forthcoming paper.

The study of type III RNases constitutes an important area in molecular biology. It is known that the pac1 + gene encodes a particular RNase III that shares low amino acid similarity with other genes despite having a double-stranded ribonuclease activity. Bioinformatics methods based on sequence alignment may fail when there is a low amino acidic identity percentage between query sequence and others with similar functions (remote homologues) or a similar sequence is not recorded in the database. Quantitative Structure-Activity Relationships (QSAR) applied to protein sequences may allow an alignment-independent prediction of protein function. These sequences QSAR like methods often use 1D sequence numerical parameters as the input to seek sequence-function relationships. However, previous 2D representation of sequences may uncover useful higher-order information. In the work described here we calculated for the first time the Spectral Moments of a Markov Matrix (MMM) associated with a 2D-HP-map of a protein sequence. We used MMMs values to characterize numerically 81 sequences of type III RNases and 133 proteins of a control group. We subsequently developed one MMM-QSAR and one classic Hidden Markov Model (HMM) based on the same data. The MMM-QSAR showed a discrimination power of RNAses from other proteins of 97.35% without using alignment, which is a result as good as for the known HMM techniques. We also report for the first time the isolation of a new Pac1 protein (DQ647826) from Schizosaccharomyces pombe, strain 428-4-1. The MMM-QSAR model predicts the new RNase III with the same accuracy as other classical alignment methods. Experimental assay of this protein confirms the predicted activity. The present results suggest that MMM-QSAR models may be used for protein function annotation avoiding sequence alignment with the same accuracy of classic HMM models.