On the Faria's inequality for the Laplacian and signless Laplacian spectra: a unified approach

Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.

Bounds for the signless laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. © 2010 Elsevier Inc. All rights reserved.

Extremal graphs for the sum of the two largest signless Laplacian eigenvalues

Let G be a simple graph on n vertices and e(G) edges. Consider the signless Laplacian, Q(G) = D + A, where A is the adjacency matrix and D is the diagonal matrix of the vertices degree of G. Let q1(G) and q2(G) be the first and the second largest eigenvalues of Q(G), respectively, and denote by S+ n the star graph with an additional edge. It is proved that inequality q1(G)+q2(G) e(G)+3 is tighter for the graph S+ n among all firefly graphs and also tighter to S+ n than to the graphs Kk _ Kn−k recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, it is conjectured that S+ n minimizes f(G) = e(G) − q1(G) − q2(G) among all graphs G on n vertices.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

On the Fucik spectrum of the Laplacian on a torus

We study the Fucik spectrum of the Laplacian on a two-dimensional torus T(2). Exploiting the invariance properties of the domain T(2) with respect to translations we obtain a good description of large parts of the spectrum. In particular, for each eigenvalue of the Laplacian we will find an explicit global curve in the Fucik spectrum which passes through this eigenvalue; these curves are ordered, and we will show that their asymptotic limits are positive. On the other hand, using a topological index based on the mentioned group invariance, we will obtain a variational characterization of global curves in the Fucik spectrum; also these curves emanate from the eigenvalues of the Laplacian, and we will show that they tend asymptotically to zero. Thus, we infer that the variational and the explicit curves cannot coincide globally, and that in fact many curve crossings must occur. We will give a bifurcation result which partially explains these phenomena. (C) 2008 Elsevier Inc. All rights reserved.

Energy of line graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. The energy of a matrix is equal to the sum of its singular values. We establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G. © 2010 Elsevier B.V. All rights reserved.

Majorantes para a ordem de subgrafos induzidos k-regulares

Muitos dos problemas de otimização em grafos reduzem-se à determinação de um subconjunto de vértices de cardinalidade máxima que induza um subgrafo k-regular. Uma vez que a determinação da ordem de um subgrafo induzido k-regular de maior ordem é, em geral, um problema NP-difícil, são deduzidos novos majorantes, a determinar em tempo polinomial, que em muitos casos constituam boas aproximações das respetivas soluções ótimas. Introduzem-se majorantes espetrais usando uma abordagem baseada em técnicas de programação convexa e estabelecem-se condições necessárias e suficientes para que sejam atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro das matrizes de adjacência, laplaciana e laplaciana sem sinal. É ainda apresentado um algoritmo não polinomial para a determinação de umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular de ordem máxima para uma classe particular de grafos. Finalmente, faz-se um estudo computacional comparativo com vários majorantes e apresentam-se algumas conclusões.

Towards a single crystal Raman spectrum of kaolinite at 77 K

The Raman spectra at 77 K of the hydroxyl stretching of kaolinite were obtained along the three axes perpendicular to the crystal faces. Raman bands were observed at 3616, 3658 and 3677 cm−1 together with a distinct band observed at 3691 cm−1 and a broad profile between 3695 and 3715 cm−1. The band at 3616 cm−1 is assigned to the inner hydroxyl. The bands at 3658 and 3677 cm−1 are attributed to the out-of-phase vibrations of the inner surface hydroxyls. The Raman spectra of the in-phase vibrations of the inner-surface hydroxyl-stretching region are described in terms of transverse and longitudinal optic splitting. The band at 3691 cm−1 is assigned to the transverse optic and the broad profile to the longitudinal optic mode. This splitting remained even at liquid nitrogen temperature. The transverse optic vibration may be curve resolved into two or three bands, which are attributed to different types of hydroxyl groups in the kaolinite.

Single crystal Raman spectroscopy of brandholzite Mg[Sb(OH)6]2•6H2O and bottinoite Ni[Sb(OH)6]2•6H2O and the polycrystalline Raman spectrum of mopungite Na[Sb(OH)6]

The single crystal Raman spectra of minerals brandholzite and bottinoite, formula M[Sb(OH)6]2•6H2O, where M is Mg+2 and Ni+2 respectively, and the non-aligned Raman spectrum of mopungite, formula Na[Sb(OH)6], are presented for the first time. The mixed metal minerals comprise of alternating layers of [Sb(OH)6]-1 octahedra and mixed [M(H2O)6]+2 / [Sb(OH)6]-1 octahedra. Mopungite comprises hydrogen bonded layers of [Sb(OH)6]-1 octahedra linked within the layer by Na+ ions. The spectra of the three minerals were dominated by the Sb-O symmetric stretch of the [Sb(OH)6]-1 octahedron, which occurs at approximately 620 cm-1. The Raman spectrum of mopungite showed many similarities to spectra of the di-octahedral minerals informing the view that the Sb octahedra gave rise to most of the Raman bands observed, particularly below 1200 cm-1. Assignments have been proposed based on the spectral comparison between the minerals, prior literature and density field theory calculations of the vibrational spectra of the free [Sb(OH)6]-1 and [M(H2O)6]+2 octahedra by a model chemistry of B3LYP/6-31G(d) and lanl2dz for the Sb atom. The single crystal data spectra showed good mode separation, allowing the majority of the bands to be assigned a symmetry species of A or E.

From slum to suburbia : examining a spectrum of planned and unplanned urban morphology in humid sub-tropical regions of the world

The current rapid urban growth throughout the world manifests in various ways and historically cities have grown, similarly, alternately or simultaneously between planned extensions and organic informal settlements (Mumford, 1989). Within cities different urban morphological regions can reveal different contexts of economic growth and/or periods of dramatic social/technological change (Whitehand, 2001, 105). Morpho-typological study of alternate contexts can present alternative models and contribute to the present discourse which questions traditional paradigms of urban planning and design (Todes et al, 2010). In this study a series of cities are examined as a preliminary exploration into the urban morphology of cities in ‘humid subtropical’ climates. From an initial set of twenty, six cities were selected: Sao Paulo, Brazil; Jacksonville, USA; Maputo, Mozambique; Kanpur, India; Hong Kong, China; and Brisbane, Australia. The urban form was analysed from satellite imagery at a constant scale. Urban morphological regions (types) were identified as those demonstrating particular consistant characteristics of form (density, typology and pattern) different to their surroundings when examined at a constant scale. This analysis was correlated against existing data and literature discussing the proliferation of two types of urban development, ‘informal settlement’ (defined here as self-organised communities identifiable but not always synonymous with ‘slums’) and ‘suburbia’ (defined here as master planned communities of generally detached houses prevalent in western society) - the extreme ends of a hypothetical spectrum from ‘planned’ to ‘spontaneous’ urban development. Preliminary results show some cities contain a wide variety of urban form ranging from the highly organic ‘self-organised’ type to the highly planned ‘master planned community’ (in the case of Sao Paulo) while others tend to fall at one end of the planning spectrum or the other (more planned in the cases of Brisbane and Jacksonville; and both highly planned and highly organic in the case of Maputo). Further research will examine the social, economical and political drivers and controls which lead to this diversity or homogeneity of urban form and speculates on the role of self-organisation as a process for the adaptation of urban form.

The 4C spectrum of fundamental behavioral relations for concurrent systems

The design of concurrent software systems, in particular process-aware information systems, involves behavioral modeling at various stages. Recently, approaches to behavioral analysis of such systems have been based on declarative abstractions defined as sets of behavioral relations. However, these relations are typically defined in an ad-hoc manner. In this paper, we address the lack of a systematic exploration of the fundamental relations that can be used to capture the behavior of concurrent systems, i.e., co-occurrence, conflict, causality, and concurrency. Besides the definition of the spectrum of behavioral relations, which we refer to as the 4C spectrum, we also show that our relations give rise to implication lattices. We further provide operationalizations of the proposed relations, starting by proposing techniques for computing relations in unlabeled systems, which are then lifted to become applicable in the context of labeled systems, i.e., systems in which state transitions have semantic annotations. Finally, we report on experimental results on efficiency of the proposed computations.

Differential pathologic variables and outcomes across the spectrum of adenocarcinoma of the esophagogastric junction

Background Adenocarcinoma of the esophagogastric junction (AEG) as described by Siewert et al. is classified as one entity in the latest (7th Edition) American Joint Cancer Committee/International Union Against Cancer (AJCC/UICC) manual, compared with the previous mix of esophageal and gastric staging systems. The origin of AEG tumors, esophageal or gastric, and their biology remain controversial, particularly for AEG type II (cardia) tumors. Methods We adapted a large prospective database (n = 520: 180 type I, 182 type II, 158 type III) to compare AEG tumors under the new TNM system Pathological variables associated with prognosis were compared (pT, pN, stage, differentiation, R status, lymphovascular invasion, perineural involvement, number of positive nodes, percent of positive nodes, and tumor length), as well as overall survival. Results Compared with AEG type I tumors, type II and type III tumors had significantly (p\0.05) more advanced pN stages, greater number and percentage of positive nodes, poorer differentiation, more radial margin involvement, and more perineural invasion. In AEG type I, 14/180 patients (8%) had[6 involved nodes (pN3), compared with 16 and 30% of patients classified type II and III, respectively. Median survival was significantly (p = 0.03) improved for type I patients (38 months) compared with those with tumors classified as type II (28 months) and type III (24 months). In multivariate analysis node positivity and pN staging but not AEG site had an impact on survival. Conclusions In this series AEG type I is associated with more favorable pathologic features and improved outcomes compared with AEG type II and III. This may reflect earlier diagnosis, but an alternative possibility, that type I may be a unique paradigm with more favorable biology, requires further study. © Société Internationale de Chirurgie 2010.

The Effective and The Evocative : A Spectrum of Creative Practice Research

In the expanding literature on creative practice research, art and design are often described as a unified field. They are bracketed together (art-and-design), referred to as interchangeable terms (art/design), and nested together, as if the practices of one domain encompass the other. However it is possible to establish substantial differences in research approaches. In this chapter we argue that core distinctions arise out of the goals of the research, intentions invested in the resulting “artefacts” (creative works, products, events), and the knowledge claims made for the research outcomes. Moreover, these fundamental differences give rise to a number of contingent attributes of the research such as the forming contexts, methodological approaches, and ways of evidencing and reporting new knowledge. We do not strictly ascribe these differences to disciplinary contexts. Rather, we use the terms effective practice research and evocative practice research to describe the spirit of the two distinctive research paradigms we identify. In short, effective practice research (often pursued in design fields) seeks a solution (or resolution) to a problem identified with a particular community, and it produces an artefact that addresses this problem by effecting change (making a situation, product or process more efficient or effective in some way). On the other hand, evocative practice research (often pursued by creative arts fields) is driven by individual pre-occupations, cultural concerns or human experience more broadly. It produces artefacts that evoke affect and resonance, and are poetically irreducible in meaning. We cite recent examples of creative research projects that illustrate the distinctions we identify. We then go on to describe projects that integrate these modes of research. In this way, we map out a creative research spectrum, with distinct poles as well as multiple hybrid possibilities. The hybrid projects we reference are not presented as evidence an undifferentiated field. Instead, we argue that they integrate research modes in deliberate, purposeful and distinctive ways: employing effective practice research methods in the production of evocative artefacts or harnessing evocative (as well as effective) research paradigms to effect change.