Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.Comment: 30 pages, 8 figure

Resistivity and Thermopower of Ni2.19Mn0.81Ga

In this paper, we report results of the first studies on the thermoelectric power (TEP) of the magnetic heusler alloy Ni$_{2.19}$Mn$_{0.81}$Ga. We explain the observed temperature dependence of the TEP in terms of the crystal field (CF) splitting and compare the observed behavior to that of the stoichiometric system Ni$_2$MnGa. The resistivity as a function of temperature of the two systems serves to define the structural transition temperature, T$_M$, which is the transition from the high temperature austenitic phase to low temperatures the martensitic phase. Occurrence of magnetic (Curie-Weiss) and the martensitic transition at almost the same temperature in Ni$_{2.19}$Mn$_{0.81}$Ga has been explained from TEP to be due to changes in the density of states (DOS) at the Fermi level.Comment: 12 pages, 4 figures, Accepted in Physical Review B vol 70, Issue 1

Pollination boilogy and breeding system of Eugenia discifera Gamble An endangered species of Western Ghats, India.

Knowledge of the pollination biology and breeding systems is very important for understanding the life history of long - lived tree species. In case of endangered species, the information may have implication for conservation practices. In the present investigation on Eugenia discifera Gamble, an endangered tree species, belongs to the family Myrtaceae, the reproductive phenology, floral biology, pollination biology and breeding system has been studied during 2012 - 2013. Based on the observation, the lifespan of a single flower was about 1-2 days and to complete the whole process of fruit formation was 102 - 112 days. Inflorescence is terminal raceme or axillary, flowers are hermaphrodite (bisexual), zygomorphic, epigynous and complete. Sepals and petals are four. Stamens are conspicuous and numerous, stigma simple, fruits are globoid with crowed by calyx lobes. Breeding experiments showed there is no apomixis. The lowest mean fruit set were observed with autogamy pollination (34.00±16.73 %) and highest mean fruit set were observed with geitonogamy pollination (48.00±21.68 %) respectively . However, there is no significant difference in the fruit set by geitonogamy and xenogamy. The flowers are often visited by various pollinating insects. The most frequent visitors are butterfly, Apis sp, Anomala sp, Altica cyanea (Weber) and they visit to collect food sources such as pollen and nectar for their survival in turn helps in pollinat ion. The factors responsible for declining the population of Eugenia discifera were recorde