On finite groups with prescribed two-generator subgroups and integral Cayley graphs

In this paper, we characterize the finite groups 𝐺 of even order with the property that, for any involution 𝑥 and element 𝑦 of 𝐺, ⟨ x , y ⟩ \langle x,y\rangle is isomorphic to one of the following groups: Z 2 \mathbb{Z}_{2} , Z 2 2 \mathbb{Z}_{2}^{2} , Z 4 \mathbb{Z}_{4} , Z 6 \mathbb{Z}_{6} , Z 2 × Z 4 \mathbb{Z}_{2}\times\mathbb{Z}_{4} , Z 2 × Z 6 \mathbb{Z}_{2}\times\mathbb{Z}_{6} and A 4 A_{4} . As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree 3 have integral spectrum.

A broadband spectral analysis of 4U 1702-429 using XMM-Newton and BeppoSAX data

Context. Most of the X-ray binary systems containing neutron stars classified as Atoll sources show two different spectral states, referred to as soft and hard. Moreover, a large number of these systems show a reflection component relativistically smeared in their spectra, which provides information on the innermost region of the system. Aims. Our aim is to investigate the poorly studied broadband spectrum of the low-mass X-ray binary system 4U 1702-429, which was recently analysed combining XMM-Newton and INTEGRAL data. The peculiar value of the reflection fraction brought us to analyse further broadband spectra of 4U 1702-429. Methods. We re-analysed the spectrum of the XMM-Newton/INTEGRAL observation of 4U 1702-429 in the 0.3–60 keV energy range and we extracted three 0.1–100 keV spectra of the source analysing three observations collected with the BeppoSAX satellite. Results. We find that the XMM-Newton/INTEGRAL spectrum is well fitted using a model composed of a disc blackbody plus a Comptonised component and a smeared reflection component. We used the same spectral model for the BeppoSAX spectra, finding that the addition of a smeared reflection component is statistically significant. The best-fit values of the parameters are compatible to each other for the BeppoSAX spectra. We find that the reflection fraction is 0.05−0.01+0.3 for the XMM-Newton/INTEGRAL spectrum and between 0.15 and 0.4 for the BeppoSAX ones. Conclusions. The relative reflection fraction and the ionisation parameter are incompatible between the XMM-Newton/INTEGRAL and the BeppoSAX observations and the characteristics of the Comptonising corona suggest that the source was in a soft state in the former observation and in a hard state in the latter.

Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

It is shown that distance powers of an integral Cayley graph over an abelian group $\Gamma$ are again integral Cayley graphs over $\Gamma$. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.