(2020) Extremal trees for the modified first Zagreb connection index with fixed number of segments or vertices of degree 2,

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The modified first Zagreb connection index ZC1∗ is a graph invariant that appeared about fifty years ago within a study of molecular modeling, and after a long time, it has been revisited in two papers ((Ali and Trinajstić, 2018) and (Naji et al., 2017)) independently. For a graph G, this graph invariant is defined as ZC1∗G=∑v∈VGdvτv, where dv is the degree of the vertex v and τv is the connection number of v (that is, the number of vertices having distance 2 from v). In this paper, the graphs with maximum/minimum ZC1∗ value are characterized from the class of all n-vertex trees with fixed number of pendent vertices (that are the vertices of degree 1).

Over the past few years, Deep-learning (DL) based modelling solutions have been presented as an alternate to the timetedious and computationally draining conventional design and optimization procedure of metasurfaces. While designing a phase-based transmission meta-device, such as meta-lenses and meta-holograms etc., the most crucial part is to optimize its unit-cell to ensure maximum electromagnetic (EM) transmission amplitude and full phase coverage (0-2π). Most of the DL-based solutions have resulted in accurate optimization to provide the desired transmission amplitude. But the abrupt discontinuities in the phase response, makes it more challenging to map and predict the optimized structure for full phase coverage. Here, we present a novel DL-based tool named as "Meta-Magus" to design transmission based metaholograms. Meta-Magus consists of two parts: (i) unit-cell optimization, and (ii) phase mask generation. Here, the first component takes target transmission amplitude, phase, material properties, and the wavelength aimed as input, process it via regression based tandem neural network, and provide optimized unit-cell structural parameters as output. Target image whose hologram is to be generated is fed to the second component which comprises of deep convolutional neural network to generate the corresponding phase mask as output. A full-wave commercial simulator then maps the optimized unit-cell onto the generated phase mask and generates the intended meta-hologram. Simulation results of the generated designs exhibit perfect holography, and validates that the model yields excellent predictions of a complete metasurface design from scratch within a matter of seconds.

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