Energy, Laplacian energy of double graphs and new families of equienergetic graphs

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \cdots, v_n\}$, the extended double cover $G^*$ is a bipartite graph with bipartition (X, Y), $X=\{x_1, x_2, \cdots, x_n\}$ and $Y=\{y_1, y_2, \cdots, y_n\}$, where two vertices $x_i$ and $y_j$ are adjacent if and only if $i=j$ or $v_i$ adjacent to $v_j$ in $G$. The double graph $D[G]$ of $G$ is a graph obtained by taking two copies of $G$ and joining each vertex in one copy with the neighbours of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs $G^*$ and $D[G]$, $L$-spectra of $G^{k*}$ the $k$-th iterated extended double cover of $G$. We obtain a formula for the number of spanning trees of $G^*$. We also obtain some new families of equienergetic and $L$-equienergetic graphs.Comment: 23 pages, 1 figur

Monitoring and testing different doses of disparlure for Indian gypsy moth, Lymantria obfuscata, in a temperate region of India (Kashmir Valley)

Pheromone traps with different doses of disparlure [(Z)-7,8-epoxy-2-methyloctadecane] were tested for a local strain of the Indian gypsy moth (Lymantria obfuscata) at Sher-e-Kashmir University of Agricultural Sciences and Technology of Kashmir (India). Disparlure at 500 µg dose proved to be effective in trapping gypsy moth populations. The first adults were caught on the third week of June in 2007-2009 with peak catches a week later. Catches in disparlure-baited traps at all dosage levels (0.5, 50 and 500 µg) were significantly higher as compared to control traps. The regression equation revealed strong (99%) correlation between moth catches and applied doses. The accumulated degree day model predicted 65.31 to 117.97 heat units for larval hatch and 794.66 to 928.15 heat units for adult emergence. The principal component analysis showed significant variability between weather variables and adult L. obfuscata population

On α\alpha-adjacency energy of graphs and Zagreb index

Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix $A_{\alpha}(G)$ of the convex combinations of $D(G)$ and $A(G)$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, for $0\leq \alpha\leq 1$. If $\rho_{1}\geq \rho_{2}\geq \dots \geq \rho_{n}$ are the eigenvalues of $A_{\alpha}(G)$ (which we call $\alpha$-adjacency eigenvalues of $G$), the $\alpha$-adjacency energy of $G$ is defined as $E^{A_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i-\frac{2\alpha m}{n}\right|$, where $n$ is the order and $m$ is the size of $G$. We obtain the upper and lower bounds for $E^{A_{\alpha}}(G)$ in terms of order $n$, size $m$ and Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we characterize the extremal graphs attaining these bounds.Comment: 17 page

Essentialist Reasoning and Knowledge Effects on Biological Reasoning in Young Children

Biological kinds undergo a variety of changes during their life span, and these changes vary in degree by organism. Understanding that an organism, such as a caterpillar, maintains category identity over its life span despite dramatic changes is a key concept in biological reasoning. At present, we know little about the developmental trajectory of children’s understanding of dramatic life-cycle changes and how this might relate to their understanding of evolution. We suggest that this understanding is a key precursor to later understanding of evolutionary change. Two studies examined the impact of age and knowledge on children’s biological reasoning about living kinds that undergo a range of natural life-span changes—from subtle to dramatic. The participants, who were 3, 4, and 7 years old, were shown paired pictures of juvenile and adult animals and asked to endorse biological or nonbiological causal mechanisms to account for life-span change. Additionally, reasoning of 3- and 4-year-old participants was compared before and after exposure to caterpillars transforming into butterflies. The results are framed in terms of a developmental trajectory in essentialist reasoning, a cognitive bias that has been associated with difficulties in understanding and accepting evolution

On Energy, Laplacian Energy and PP-fold Graphs

For a graph $G$ having adjacency spectrum ($A$-spectrum) $\lambda_n\leq\lambda_{n-1}\leq\cdots\leq\lambda_1$ and Laplacian spectrum ($L$-spectrum) $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$, the energy is defined as $E(G)=\sum_{i=1}^{n}|\lambda_i|$ and the Laplacian energy is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\frac{2m}{n}|$. In this paper, we give upper and lower bounds for the energy of $KK_n^j,~1\leq j \leq n$ and as a consequence we generalize a result of Stevanovic et al. [More on the relation between Energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. {\bf 61} (2009) 395-401]. We also consider strong double graph and strong $p$-fold graph to construct some new families of graphs $G$ for which E(G)> LE(G)

Genetic diversity studies in common bean (Phaseolus vulgaris L.) using molecular markers

Molecular characterization of thirteen common bean genotypes was done with random amplified polymorphic DNA (RAPD) markers. Initially, 15 primers were screened out of which only seven were selected which generated a total of 65 amplification products out of which 63 bands (96.62%) were polymorphic indicating fair amount of polymorphism. The genotypes shared 43% genetic similarity among themselves. Cluster analysis delineated the genotypes into three groups with seven, five and one genotype in cluster-I, II and III, respectively. The maximum similarity index (82.35) based dice similarity coefficient was obtained between SKUA-R-21 and SKUA-R-19, while it was minimum (27.72) between genotypes PBG-29 and SKUA-R-01.Key words: Genetic divergence, common bean, random amplified polymorphic DNA (RAPD)