Connectivity of Direct Products of Graphs

Let $\kappa(G)$ be the connectivity of $G$ and $G\times H$ the direct product of $G$ and $H$. We prove that for any graphs $G$ and $K_n$ with $n\ge 3$, $\kappa(G\times K_n)=min\{n\kappa(G),(n-1)\delta(G)\}$, which was conjectured by Guji and Vumar.Comment: 5 pages, accepted by Ars Com

From Equivalence Principles to Cosmology: Cosmic Polarization Rotation, CMB Observation, Neutrino Number Asymmetry, Lorentz Invariance and CPT

In this paper, we review the approach leading to cosmic polarization rotation observation and present the current status with an outlook. In the study of the relations among equivalence principles, we found that long-range pseudoscalar-photon interaction is allowed. Pseudoscalar-photon interaction would induce a rotation of linear polarization of electromagnetic wave propagating with cosmological/astrophysical distance. In 2002, DASI successfully observed the polarization of the cosmological microwave background radiation. In 2003, WMAP observed the correlation of polarization with temperature anisotropy at more than 10 sigma in the cosmological microwave background. From this high polarization-temperature correlation in WMAP observation, we put a limit of 0.1 rad on the rotation of linear polarization of cosmological microwave background (CMB) propagation. Pseudoscalar-photon interaction is proportional to the gradient of the pseudoscalar field. From phenomenological point of view, this gradient could be neutrino number asymmetry current, other density current, or a constant vector. In these situations, Lorentz invariance or CPT may or may not effectively be violated. In this paper, we review and compile various results. Better accuracy in CMB polarization observation is expected from PLANCK mission to be launched next year. A dedicated CMB polarization observer in the future would probe this fundamental issue more deeply.Comment: 9 pages, 2 figures, a few references with corresponding text change added in this version, invited talk given in VIII Asia-Pacific International Conference on Gravitation and Astophysics (ICGA8), August 29 - September 1, 2007, Nara Women's University, Japan, submitted to Progress of Theoretical Physics Supplemen

Gravitational Wave (GW) Classification, Space GW Detection Sensitivities and AMIGO (Astrodynamical Middle-frequency Interferometric GW Observatory)

After first reviewing the gravitational wave (GW) spectral classification. we discuss the sensitivities of GW detection in space aimed at low frequency band (100 nHz-100 mHz) and middle frequency band (100 mHz-10 Hz). The science goals are to detect GWs from (i) Supermassive Black Holes; (ii) Extreme-Mass-Ratio Black Hole Inspirals; (iii) Intermediate-Mass Black Holes; (iv) Galactic Compact Binaries; (v) Stellar-Size Black Hole Binaries; and (vi) Relic GW Background. The detector proposals have arm length ranging from 100 km to 1.35x109 km (9 AU) including (a) Solar orbiting detectors and (b) Earth orbiting detectors. We discuss especially the sensitivities in the frequency band 0.1-10 microHz and the middle frequency band (0.1 Hz-10 Hz). We propose and discuss AMIGO as an Astrodynamical Middle-frequency Interferometric GW Observatory.Comment: 10 pages, 2 figures, 1 table, Plenary talk given in Joint Meeting of 13th International Conference on Gravitation, Astrophysics, and Cosmology and 15th Italian-Korean Symposium on Relativistic Astrophysics, Ewha Womans University, Seoul, Korea, July 3-7, 201