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A proper Skolem labelling of a graph G is a function assigning a positive integer to each vertex of G such that any two vertices assigned the same integer are that distance apart in the graph. The Skolem number of a graph is smallest number n such that there exists a proper Skolem labelling only using the positive integers less than or equal to n. In this paper, we will begin by proving the Skolem number for another family of subgraphs of the hexagonal lattice and then prove the Skolem number for two families of subgraphs of the Kagome Lattice. © 2025 Georgia Southern University. All rights reserved.
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A deep learning (DL) algorithm is built and tested for its ability to determine centers of star images in HST/WFPC2 exposures, in filters F555W and F814W. These archival observations hold great potential for proper-motion studies, but the undersampling in the camera’s detectors presents challenges for conventional centering algorithms. Two exquisite data sets of over 600 exposures of the cluster NGC 104 in these filters are used as a testbed for training and evaluating the DL code. Results indicate a single-measurement standard error from 8.5 to 11 mpix, depending on the detector and filter. This compares favorably to the ∼20 mpix achieved with the customary “effective point spread function (PSF)” centering procedure for WFPC2 images. Importantly, the pixel-phase error is largely eliminated when using the DL method. The current tests are limited to the central portion of each detector; in future studies, the DL code will be modified to allow for the known variation of the PSF across the detectors.
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We measure the absolute proper motions of Andromeda V (And V) and Andromeda VI/Pegasus (And VI) dwarf galaxies, satellites of M31 located near its galactic plane. And VI is located the farthest from M31 among the six satellites with currently measured proper motions. A combination of Advanced Camera for Surveys/wide filed channel (WFC) and WFPC2 exposures is utilized, spanning a 20 yr time baseline. The WFPC2 exposures are processed using a recently developed deep-learning centering procedure as well as the most up-to-date astrometric calibration of the camera. We use on the order of 100 background galaxies per satellite to determine the correction to absolute proper motion. For And V, we obtain an absolute proper motion of (μα, μδ)And V= (26.1 ± 21.5 , − 74.2 ± 19.1) μ as yr−1. For And VI, we obtain an absolute proper motion of (μα, μδ)And VI= (− 1.6 ± 12.3 , − 52.6 ± 11.2) μ as yr−1. Orbit integrations and analyses are made for these two Andromeda satellites using two estimates of both the mass and proper motion of M31. It is found that And V has an orbit consistent within errors with alignment with M31’s disk and counter orbiting it, although this alignment is not well constrained. And VI’s orbit is better determined and is very much consistent with coorbiting with M31’s disk. While currently at a distance of ∼280 kpc from M31, And VI will remain beyond a distance of ∼90 kpc from M31, thus experiencing low tidal influence compared to the other M31 satellites with known orbits. Both satellites are determined to be well-bound to M31. © 2025. The Author(s). Published by the American Astronomical Society.
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We measure the absolute proper motion of Andromeda III (And III) using Advanced Camera for Surveys/Wide Field Channel and WFPC2 exposures spanning an unprecedented 22 yr time baseline. The WFPC2 exposures have been processed using a deep-learning centering procedure recently developed as well as an improved astrometric calibration of the camera. The absolute proper motion zero point is given by 98 galaxies and 16 Gaia EDR3 stars. The resulting proper motion is (μ α , μ δ ) = (−10.5 ± 12.5, 47.5 ± 12.5) μas yr−1. We perform an orbit analysis of And III using two estimates of M31's mass and proper motion. We find that And III’s orbit is consistent with dynamical membership to the Great Plane of Andromeda system of satellites although with some looser alignment compared to the previous two satellites NGC 147 and NGC 185. And III is bound to M31 if M31's mass is M vir ≥ 1.5 × 1012 M ⊙.
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- English (4)