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.

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number eta(G) of a graph G is the largest integer n for which the complete graph K-n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, eta(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G square H of graphs. As the main result of this paper, we prove that eta(G(1) square G(2)) >= h root 1 (1 - o(1)) for any two graphs G(1) and G(2) with eta(G(1)) = h and eta(G(2)) = l. We show that the above lower bound is asymptotically best possible when h >= l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G(1) square G(2) square ... square G(k) be the ( unique) prime factorization of G. Then G satisfies Hadwiger's conjecture if k >= 2 log log chi(G) + c', where c' is a constant. This improves the 2 log chi(G) + 3 bound in [2] 2. Let G(1) and G(2) be two graphs such that chi(G1) >= chi(G2) >= clog(1.5)(chi(G(1))), where c is a constant. Then G1 square G2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G(d) (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan [2]. ( They had shown that the Hadiwger's conjecture is true for G(d) if d >= 3).

The Wiener, Eccentric Connectivity and Zagreb Indices of the Hierarchical Product of Graphs

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs having a distinguished or root vertex, labeled 0. The hierarchical product G2 ⊓ G1 of G2 and G1 is a graph with vertex set V2 × V1. Two vertices y2y1 and x2x1 are adjacent if and only if y1x1 ∈ E1 and y2 = x2; or y2x2 ∈ E2 and y1 = x1 = 0. In this paper, the Wiener, eccentric connectivity and Zagreb indices of this new operation of graphs are computed. As an application, these topological indices for a class of alkanes are computed. ACM Computing Classification System (1998): G.2.2, G.2.3.

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R-1 x R-2 x...x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a left perpendicular1 + 1/c log n right perpendicular(d-1) approximation ratio for any constant c >= 1 when d >= 2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in left perpendicular(Delta + 2) ln nright perpendicular dimensions, where Delta is the maximum degree of G. This algorithm implies that box(G) <= left perpendicular(Delta + 2) ln nright perpendicular for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Delta, we show that for almost all graphs on n vertices, their boxicity is O(d(av) ln n) where d(av) is the average degree.

Fair Sets of Some Classes of Graphs

Given a non empty set S of vertices of a graph, the partiality of a vertex with respect to S is the di erence between maximum and minimum of the distances of the vertex to the vertices of S. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of Kn, Km;n, Kn e, wheel graphs, odd cycles and symmetric even graphs are identi ed. The fair sets of the Cartesian product graphs are also discussed

Applying brand equity to the social product of breastfeeding

Social marketing has successfully adopted many of the techniques of commercial marketing; however, a key commercial marketing theory that does not appear to be utilised in social marketing theory is brand equity. Given that a key outcome of brand equity is loyalty, which is also a desired outcome of many social marketing programs, brand equity appears to be a relevant theoretical framework. This study presents descriptive results of the brand equity levels of 296 Gen Y Australians for the social product of breastfeeding. Breastfeeding is a desirable health behaviour with significant health and wellbeing outcomes for infants, mothers and communities. It was selected as the focus of this paper because loyalty to the behaviour is not increasing, according to the targets set by national government authorities.

The product of text and 'Other' statements : discourse analysis and the critical use of Foucault

Much has been written on Michel Foucault’s reluctance to clearly delineate a research method, particularly with respect to genealogy (Harwood 2000; Meadmore, Hatcher, & McWilliam 2000; Tamboukou 1999). Foucault (1994, p. 288) himself disliked prescription stating, “I take care not to dictate how things should be” and wrote provocatively to disrupt equilibrium and certainty, so that “all those who speak for others or to others” no longer know what to do. It is doubtful, however, that Foucault ever intended for researchers to be stricken by that malaise to the point of being unwilling to make an intellectual commitment to methodological possibilities. Taking criticism of “Foucauldian” discourse analysis as a convenient point of departure to discuss the objectives of poststructural analyses of language, this paper develops what might be called a discursive analytic; a methodological plan to approach the analysis of discourses through the location of statements that function with constitutive effects.

On the probability density function of the product of Rayleigh distributed random variables

In this paper we investigate the distribution of the product of Rayleigh distributed random variables. Considering the Mellin-Barnes inversion formula and using the saddle point approach we obtain an upper bound for the product distribution. The accuracy of this tail-approximation increases as the number of random variables in the product increase.

The product of text and “Other” statements : discourse analysis and the critical use of Foucault

Much has been written on Michel Foucault’s reluctance to clearly delineate a research method, particularly with respect to genealogy (Harwood 2000; Meadmore, Hatcher, & McWilliam 2000; Tamboukou 1999). Foucault (1994, p. 288) himself disliked prescription stating, “I take care not to dictate how things should be” and wrote provocatively to disrupt equilibrium and certainty, so that “all those who speak for others or to others” no longer know what to do. It is doubtful, however, that Foucault ever intended for researchers to be stricken by that malaise to the point of being unwilling to make an intellectual commitment to methodological possibilities. Taking criticism of “Foucauldian” discourse analysis as a convenient point of departure to discuss the objectives of poststructural analyses of language, this paper develops what might be called a discursive analytic; a methodological plan to approach the analysis of discourses through the location of statements that function with constitutive effects.

Connected learning in the library as a product of hacking, making, social diversity and messiness

Learning is most effective when intrinsically motivated through personal interest, and situated in a supportive socio-cultural context. This paper reports on findings from a study that explored implications for design of interactive learning environments through 18 months of ethnographic observations of people’s interactions at “Hack The Evening” (HTE). HTE is a meetup group initiated at the State Library of Queensland in Brisbane, Australia, and dedicated to provide visitors with opportunities for connected learning in relation to hacking, making and do-it-yourself technology. The results provide insights into factors that contributed to HTE as a social, interactive and participatory environment for learning – knowledge is created and co-created through uncoordinated interactions among participants that come from a diversity of backgrounds, skills and areas of expertise. The insights also reveal challenges and barriers that the HTE group faced in regards to connected learning. Four dimensions of design opportunities are presented to overcome those challenges and barriers towards improving connected learning in library buildings and other free-choice learning environments that seek to embody a more interactive and participatory culture among their users. The insights are relevant for librarians as well as designers, managers and decision makers of other interactive and free-choice learning environments.

The M(r) 43K major capsid protein of rice ragged stunt oryzavirus is a post-translationally processed product of a M(r) 67 348 polypeptide encoded by genome segment 8

The nucleotide sequence of DNA complementary to rice ragged stunt oryzavirus (RRSV) genome segment 8 (S8) of an isolate from Thailand was determined. RRSV S8 is 1 914 bp in size and contains a single large open reading frame (ORF) spanning nucleotides 23 to 1 810 which is capable of encoding a protein of M(r) 67 348. The N-terminal amino acid sequence of a ~43K virion polypeptide matched to that inferred for an internal region of the S8 coding sequence. These data suggest that the 43K protein is encoded by S8 and is derived by a proteolytic cleavage. Predicted polypeptide sizes from this possible cleavage of S8 protein are 26K and 42K. Polyclonal antibodies raised against a maltose binding protein (MBP)-S8 fusion polypeptide (expressed in Escherichia coli) recognised four RRSV particle associated polypeptides of M(r) 67K, 46K, 43K and 26K and all except the 26K polypeptide were also highly immunoreactive to polyclonal antibodies raised against purified RRSV particles. Cleavage of the MBP-S8 fusion polypeptide with protease Factor X produced the expected 40K MBP and two polypeptides of apparent M(r) 46K and 26K. Antibodies to purified RRSV particles reacted strongly with the intact fusion protein and the 46K cleavage product but weakly to the 26K product. Furthermore, in vitro transcription and translation of the S8 coding region revealed a post-translational self cleavage of the 67K polypeptide to 46K and 26K products. These data indicate that S8 encodes a structural polypeptide, the majority of which is auto- catalytically cleaved to 26K and 46K proteins. The data also suggest that the 26K protein is the self cleaving protease and that the 46K product is further processed or undergoes stable conformational changes to a ~43K major capsid protein.

A product of their culture : Using a value systems approach to understand the work practices of journalists

Past approaches adopted by scholars in comparing international news have tended to concentrate on political and economic perspectives, while the role that culture plays in determining news has been somewhat neglected until recently. This article examines the role of culture in the development of journalistic practices and how a value systems approach can be applied to understanding journalism practices across cultures. Specifically, the article compares German and Anglo-American journalism practices with a view to locating differences between these traditions. The study demonstrates that using value systems as developed by Dutch anthropologist Geert Hofstede can be immensely useful in comparing the differences between the two traditions, as well as in understanding how journalists in these traditions report about the world.

Sport development programmes for Indigenous Australians : innovation, inclusion and development, or a product of ‘white guilt’?

Under the legacy of neoliberalism, it is important to consider how the indigenous people, in this case of Australia, are to advance, develop and achieve some approximation of parity with broader societies in terms of health, educational outcomes and economic participation. In this paper, we explore the relationships between welfare dependency, individualism, responsibility, rights, liberty and the role of the state in the provision of Government-funded programmes of sport to Indigenous communities. We consider whether such programmes are a product of ‘white guilt’ and therefore encourage dependency and weaken the capacity for independence within communities and individuals, or whether programmes to increase rates of participation in sport are better viewed as good investments to bring about changes in physical activity as (albeit a small) part of a broader social policy aimed at reducing the gaps between Indigenous and non-Indigenous Australians in health, education and employment.

A reply to a discussion by D.J. Cook of the paper a reaction product of lime and silica from rice husk asha star, open

Abstract is not available.