Switching Equivalence in Symmetric n-Sigraphs-V

Introducing a new notion S-antipodal symmetric n-sigraph of a symmetric n-sigraph and its properties are obtained. Also giving the relation between antipodal symmetric n-sigraphs and S-antipodal symmetric n-sigraphs. Further, discussing structural characterization of S-antipodal symmetric n-sigraphs

A Note On Jump Symmetric n-Sigraph

For standard terminology and notion in graph theory we refer the reader to West; the nonstandard will be given in this paper as and when required. We treat only finite simple graphs without self loops and isolates

Crystal Structure of 2′,3′-Di-O-Acetyl-5′-Deoxy-5-Fluorocytidine with N–H···(O,F) Proton Donor Bifurcated and (C,N)–H···O Bifurcated Acceptor Dual Three-Center Hydrogen Bond Configurations

The title compound, C13H16O6N3F, features a central furan ring containing four carbon atom chiral centers with a 4-amino-5-fluoro-2-oxopyrimidine group, two acetyl groups and a methyl group bonded at the 2,3,4,5 positions, each in an absolute R configuration (2R,3R,4R,5R). It crystallizes in the monoclinic space group C2 with unit cell parameters a = 14.5341(3), b = 7.26230(10), c = 16.2197(3) Å, β = 116.607(2)°, Z = 4. An extensive array of intra and inter molecular hydrogen bond interactions dominate crystal packing in the unit cell highlighted by a relatively rare three-center proton-bifurcated donor N–H···(O,F) hydrogen bond interaction in cooperation with a second, (C,N)–H···O bifurcated acceptor three-center hydrogen bond in a supportive fashion. Additional weak Cg π-ring inter molecular interactions between a fluorine atom and the 4-amino-5-fluoro-2-oxopyrimidine ring in concert with multiple donor and acceptor hydrogen bonds significantly influence the bond distances, bond angles and torsion angles of the deoxy-5-fluorocytidine group. Comparison to a MOPAC computational calculation provides support to these observations

International Journal of Mathematical Combinatorics, Vol.6

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Sign-Compatibility of Some Derived Signed Graphs

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V, E), called the underlying graph of S and σ : E → {+1, −1} is a function from the edge set E of Su into the set {+1, −1}, called the signature of S. A sigraph S is sign-compatible if there exists a marking µ of its vertices such that the end vertices of every negative edge receive ‘−1’ marks in µ and no positive edge does so. In this paper, we characterize S such that its ×-line sigraphs, semi-total line sigraphs, semi-total point sigraphs and total sigraphs are sign-compatible

On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page

Non-topological persistence for data analysis and machine learning

This thesis main objective is to study possible applications of the generalisation of persistence theory introduced in [1], [2]. This generalisation extends the notion of persistence to a wider categorical setting, avoiding constructing secondary structures as topological spaces. The first field analysed is graph theory. At first, we studied which classical graph theory invariants could be used as rank function. Another aspect analysed in this thesis is the extension of the study of connectivity in graphs from a persistence viewpoint started in [1] to oriented graphs. Moreover, we studied how different orientation of the same underlying graph can change the distribution of cornerpoints in persistence diagrams, both in deterministic and random graphs. The other application field analysed is image processing. We adapted the notion of steady and ranging sets to the category of sets and used them to define activation and deactivation rules for each pixel. These notions allowed us to define a filter capable of enhancing the signal of pixels close to a border. This filter has proven to be stable under salt and pepper noise perturbation. At last, we used this filter to define a novel pooling layer for convolutional neural networks. In the experimental part, we compared the proposed layer with other state-of-the-art layers. The results show how the proposed layer outperform the other layers in term of accuracy. Moreover, by concatenating the proposed and the Max pooling, it is possible to improve accuracy further. [1] Bergomi, M.G., Ferri, M., Vertechi, P., Zuffi, L. (2020), Beyond topological persistence: Starting from networks, arXiv. [2] Bergomi, M. G., & Vertechi, P. (2020). Rank-based persistence. Theory and Applications of Categories, 35, 228-260