for a graph $g$ with edge set $e(g)$, the multiplicative sum zagreb index of $g$ is defined as$pi^*(g)=pi_{uvin e(g)}[d_g(u)+d_g(v)]$, where $d_g(v)$ is the degree of vertex $v$ in $g$.in this paper, we first introduce some graph transformations that decreasethis index. in application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum zagreb ...

We give a bi-criteria approximation algorithm for the Minimum Nonuniform Partitioning problem, recently introduced by Krauthgamer, Naor, Schwartz and Talwar (2014). In this problem, we are given a graph G = (V,E) on n vertices and k numbers ρ1, . . . , ρk. The goal is to partition the graph into k disjoint sets P1, . . . , Pk satisfying |Pi| ≤ ρin so as to minimize the number of edges cut by th...

background: in this study, we aimed to investigate the ability of transvaginal doppler ultrasonography in predicting in vitro fertilization (ivf) outcome in women undergoing this procedure. materials and methods: in this prospective observational study, 65 infertile women underwent ivf procedure in isfahan, during 2013-2014, were studied. the pulsatility index (pi), resistive index (ri) and pea...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

Using a recently proposed gauge covariant diagonalization of $\pi a_1$-mixing we show that the low energy theorem $F^{\pi}=e f_\pi^2 F^{3\pi}$ current algebra, relating anomalous form factor $F_{\gamma \to \pi^+\pi^0\pi^-} = and neutral pion $F_{\pi^0 \gamma\gamma}=F^\pi$, is fulfilled in framework Nambu-Jona-Lasinio (NJL) model, solving long standing problem encountered extension including vec...

the concept of geometric-arithmetic indices was introduced in the chemical graph theory.these indices are defined by the following general formula:   ( )2( ) uv e g u vu vq qq qga g ,where qu is some quantity that in a unique manner can be associated with the vertex u ofgraph g. in this paper the exact formula for two types of geometric-arithmetic index of vphenylenicnanotube are given.

A lack of diversity and durability of resistant soybean varieties complicates management of the soybean cyst nematode (SCN), Heterodera glycines, exemplified by the current overdependence on the PI 88788 source of resistance. Of interest is the effect of adaptation of a SCN population to a source of resistance on its subsequent ability to develop on others. Female indices (FI) from virulence as...

A water oxidation catalyst generated via electrodeposition from aqueous solutions containing phosphate and Co(2+) (Co-Pi) has been studied by in situ X-ray absorption spectroscopy. Spectra were obtained for Co-Pi films of two different thicknesses at an applied potential supporting water oxidation catalysis and at open circuit. Extended X-ray absorption fine structure (EXAFS) spectra indicate t...

For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...