Miscellaneous properties of embeddings of line, total and middle graphs

Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that Full-size image (<1 K) is planar, where Full-size image (<1 K) is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not Full-size image (<1 K) is planar in an Full-size image (<1 K) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs

An ultrahot Neptune in the Neptune desert

About 1 out of 200 Sun-like stars has a planet with an orbital period shorter than one day: an ultrashort-period planet. All of the previously known ultrashort-period planets are either hot Jupiters, with sizes above 10 Earth radii (R⊕), or apparently rocky planets smaller than 2 R⊕. Such lack of planets of intermediate size (the ‘hot Neptune desert’) has been interpreted as the inability of low-mass planets to retain any hydrogen/helium (H/He) envelope in the face of strong stellar irradiation. Here we report the discovery of an ultrashort-period planet with a radius of 4.6 R⊕ and a mass of 29 M⊕, firmly in the hot Neptune desert. Data from the Transiting Exoplanet Survey Satellite revealed transits of the bright Sun-like star LTT 9779 every 0.79 days. The planet’s mean density is similar to that of Neptune, and according to thermal evolution models, it has a H/He-rich envelope constituting 9.0^(+2.7)_(−2.9)% of the total mass. With an equilibrium temperature around 2,000 K, it is unclear how this ‘ultrahot Neptune’ managed to retain such an envelope. Follow-up observations of the planet’s atmosphere to better understand its origin and physical nature will be facilitated by the star’s brightness (V_(mag) = 9.8)

An ultrahot Neptune in the Neptune desert

About 1 out of 200 Sun-like stars has a planet with an orbital period shorter than one day: an ultrashort-period planet1,2. All of the previously known ultrashort-period planets are either hot Jupiters, with sizes above 10 Earth radii (R⊕), or apparently rocky planets smaller than 2 R⊕. Such lack of planets of intermediate size (the ‘hot Neptune desert’) has been interpreted as the inability of low-mass planets to retain any hydrogen/helium (H/He) envelope in the face of strong stellar irradiation. Here we report the discovery of an ultrashort-period planet with a radius of 4.6 R⊕ and a mass of 29 M⊕, firmly in the hot Neptune desert. Data from the Transiting Exoplanet Survey Satellite3 revealed transits of the bright Sun-like star LTT 9779 every 0.79 days. The planet’s mean density is similar to that of Neptune, and according to thermal evolution models, it has a H/He-rich envelope constituting 9.0+2.7−2.9% of the total mass. With an equilibrium temperature around 2,000 K, it is unclear how this ‘ultrahot Neptune’ managed to retain such an envelope. Follow-up observations of the planet’s atmosphere to better understand its origin and physical nature will be facilitated by the star’s brightness (Vmag = 9.8).<br

Make EU trade with Brazil sustainable

Brazil, home to one of the planet's last great forests, is currently in trade negotiations with its second largest trading partner, the European Union (EU). We urge the EU to seize this critical opportunity to ensure that Brazil protects human rights and the environment