Vol. 56 No. 3 (259) (2022)

DOI: https://doi.org/10.46991/PYSUA.2022.56.3
Published: 2022-10-17

Mathematics

  • Mathematics

    ON THE PALETTE INDEX OF GRAPHS HAVING A SPANNING STAR

    Aghasi B. Ghazaryan, Petros A. Petrosyan
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    Abstract

    A proper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that $\alpha(e)\not=\alpha(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. In a proper edge coloring of a graph $G$, the palette of a vertex $v \in V(G)$ is the set of colors assigned to the edges incident to $v$. The palette index of $G$ is the minimum number of distinct palettes occurring in $G$ among all proper edge colorings of $G$. A graph $G$ has a spanning star, if it has a spanning subgraph which is a star. In this paper, we consider the palette index of graphs having a spanning star. In particular, we give sharp upper and lower bounds on the palette index of these graphs. We also provide some upper and lower bounds on the palette index of the complete split and threshold graphs.

    References

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    West D.B. Introduction to Graph Theory. Prentice Hall (2001), 588. https://books.google.am/books?id=TuvuAAAAMAAJ

    Horňák M., Kalinowski R., et al. Minimum Number of Palettes in Edge Colorings. Graph. Comb. 30 (2014), 619-626. https://doi.org/10.1007/s00373-013-1298-8

    Bonvicini S., Mazzuoccolo G. Edge-Colorings of 4-regular Graphs with the Minimum Number of Palettes. Graph. Comb. 32 (2016), 1293-1311. https://doi.org/10.1007/s00373-015-1658-7

    Horňák M., Hudák J. On the Palette Index of Complete Bipartite Graphs. Discuss. Math. Graph Theory 38 (2017), 463--476. https://doi.org/10.7151/dmgt.2015

    Casselgren C.J., Petrosyan P.A. Some Results on the Palette Index of Graphs. Discret. Math. Theor. Comput. Sci. 21 (2019). https://doi.org/10.23638/DMTCS-21-3-11

    Bonisoli A., Bonvicini S., Mazzuoccolo G. On the Palette Index of a Graph: the Case of Trees. Lecture Notes of Seminario Interdisciplinare di Matematica 14 (2017), 49-55. http://hdl.handle.net/11380/1132584

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    Avesani M., Bonisoli A., Mazzuoccolo G. A Family of Multigraphs with Large Palette Index. ARS Mathematica Contemporanea 17 (2019), 115-124. https://doi.org/10.26493/1855-3974.1528.d41

    Mattiolo D., Mazzuoccolo G., Tabarelli G. Graphs with Large Palette Index. Discrete Math. 345 (2022), 112814. https://doi.org/10.1016/j.disc.2022.112814

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  • Mathematics

    ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH n-INDEPENDENT NODES

    Hakop A. Hakopian
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    Abstract

    Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e. each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three linearly independent curves of degree less than or equal to $n-1$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: either all its nodes belong to a curve of degree $n-2,$ or all its nodes but three belong to a (maximal) curve of degree $n-3.$ This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. Hakopian. Note that the proofs of the two results are completely different.

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    Rafayelyan L. Poised Nodes Set Constructions on Algebraic Curves. East J. Approx. 17 (2011), 285-298.

    Hakopian H., Toroyan S. On the Uniqueness of Algebraic Curves Passing Through $n$-independent Nodes. New York J. Math. 22 (2016), 441-452.

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    Hakopian H., Kloyan H., Voskanyan D. On plane Algebraic Curves Passing Through $n$-independent Nodes. J. Cont. Math. Anal. 56 (2021), 280-294. https://doi.org/10.48550/arXiv.2105.13863

    Hakopian H., Kloyan H. On the Dimension of Spaces of Algebraic Curves Passing Through $n$-independent nodes. Proceedings of the YSU. Phys. and Math. Sci. 53 (2019), 91-100. https://doi.org/10.46991/PYSU:A/2019.53.2.091

  • Mathematics

    ON REGULAR PARAMEDIAL DIVISION ALGEBRAS

    David N. Harutyunyan
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    Abstract

    In this paper $n$-ary regular division algebras are discussed, which are  satisfying the hyperidentity of paramediality. It is shown that every operation  in $n$-ary regular paramedial division algebra will be linearly represented over the same Abelian group. Similar results already obtained for regular medial division algebras in [1].

    References

    Davidov S.S. On Regular Medial Division Algebras. Quasigr. Relat. Syst. 21 (2013), 155--164.

    Movsisyan Yu. Hyperidentities: Boolean and De Morgan Structures. World Scientific (2022), 560 p. https://doi.org/10.1142/12796

    Movsisyan Yu. Introduction to the Theory of Algebras with Hyperidentities. Yerevan, YSU Press (1986) (in Russian).

    Movsisyan Yu. Hyperidentities and Hypervarieties in Algebras. Yerevan, YSU Press (1990) (in Russian).

    Ehsani A., Movsisyan Yu. A Representation of Paramedial $n$-ary Groupoids. Asian-Eur. J. Math. 7 (2014), 1450020. https://doi.org/10.1142/S179355711450020X

    Davidov S. On Paramedial Division Groupoids. Asian-Eur. J. Math. 9 (2016), 1650008. https://doi.org/10.1142/S179355711650008X

Physics

  • Physics

    OPTICAL ABSORPTION IN SEMICONDUCTOR NANOWIRE MEDIATED BY ELECTRON-POLAR OPTICAL PHONON AND SPIN-ORBIT INTERACTIONS

    Tigran K. Ghukasyan
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    Abstract

    The intrasubband and intersubband absorption of light by free charge carriers in a semiconductor nanowire upon scattering by polar optical phonons mediated by Rashba and Dresselhaus spin-orbit interactions has been studied. The dependence of the absorption coefficient on the energy of the incident photon is investigated by counting the transitions between different conduction subbands. It is shown that the spin-orbit interaction leads to an increase in the coefficient of intrasubband and intersubband absorption of light, with the peaks of the absorption coefficient being determined by the energies of the absorbed photon and the absorbed or emitted phonon. In this case, the difference between the values of the absorption coefficients with or without spin-orbit interaction has maximum in the range of the local minimum of the absorption coefficient obtained by ignoring the spin-orbit coupling.

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