Vol. 59 No. 1 (266) (2025)

DOI: https://doi.org/10.46991/PYSUA.2025.59.1
Published: 2025-04-30

Mathematics

  • Mathematics

    ON STRONG CHROMATIC INDEX OF SOME OPERATIONS ON GRAPHS

    Aram K. Drambyan
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    Abstract

    A strong edge-coloring of a graph $G$ is a mapping $\phi : E(G) \rightarrow \mathbb{N}$ such that the edges at distance $0$ or $1$ receive distinct colors. The minimum number of colors required for such a coloring is called the strong chromatic index of $G$ and is denoted by $\chi_s'(G)$. In this paper, we investigate the strong chromatic index of the Mycielskian $\mu(G)$ of graphs $G$ and corona products $G \odot H$ of graphs $G$ and $H$. In particular, we give tight lower and upper bounds on $\chi_s'(G \odot H)$. Moreover, we provide specific structural criteria, under which the upper bound is sharp. We also derive tight lower and upper bounds on $\chi_s'(\mu(G))$ for Mycielskian of graphs.

    References

    West D.B. Introduction to Graph Theory. Prentice-Hall, New Jersey (2001). https://dwest.web.illinois.edu/igt/

    Fouquet J.L., Jolivet J.L. Strong Edge-colorings of Graphs and Applications to Multi-k-Gons. Ars Combinatoria 16(A) (1983), 141-150.

    Andersen L.D. The Strong Chromatic Index of a Cubic Graph is at Most 10. Discrete Math. 108 (1992), 231-252. https://doi.org/10.1016/0012-365X(92)90678-9

    Hor'ak P., Qing H., Trotter W.T. Induced Matchings in Cubic Graphs. J. Graph Theory 17 (1993), 151-160. https://doi.org/10.1002/jgt.3190170204

    Cranston D.W. Strong Edge-coloring of Graphs with Maximum Degree 4 Using 22 Colors. Discrete Math. 306 (2006), 2772-2778. https://doi.org/10.1016/j.disc.2006.03.053

    Huang M., Santana M., Yu G. Strong Chromatic Index of Graphs with Maximum Degree Four. Electron. J. Comb. 25 (2018), 3-31. https://doi.org/10.37236/7016

    Chung F.R.K., Gy'arf'as A., Trotter W.T., Tuza Z. The Maximum Number of Edges in 2K_2-free Graphs of Bounded Degree. Discrete Math. 81 (1990), 129-135. https://doi.org/10.1016/0012-365X(90)90144-7

    Bruhn H., Joos F. A Stronger Bound for the Strong Chromatic Index. Electron. Notes Discrete Math. 49 (2015), 277-284. https://doi.org/10.1016/j.endm.2015.06.038

    Eoin H., Rémi de Joannis de Verclos, Kang R.J. An Improved Procedure for Colouring Graphs of Bounded Local Density. In: Proc. of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), Society for Industrial and Applied Mathematics (2020), 135-148. https://doi.org/10.19086/aic.2022.7

    Frucht R.W., Harary F. On the Corona of Two Graphs. Aequ. Math. 4 (1970), 322-325. https://doi.org/10.1007/BF01844162

    Rohan Sh., Adhikari B., Mishra A. Structural and Spectral Properties of Coronagraphs. Discrete Appl. Math. 228 (2017), 14-31. https://doi.org/10.1016/j.dam.2017.01.005

    Arumugam S., Lee Y.C., Premalatha K., Wang T.M. On Local Antimagic Vertex Coloring for Corona Products of Graphs. arXiv preprint:1808.04956 (2018). https://doi.org/10.48550/arXiv.1808.04956

    Mycielski J. Sur Le Coloriage Des Graphs. In: Colloquium Mathematicae 3 (1955), 161-162. http://eudml.org/doc/210000

    Rudnicki P., Stewart L. The Mycielskian of a Graph. Formalized Mathematics 19 (2011), 27-34. http://eudml.org/doc/267007

    Miškuf J., Škrekovski R., Tancer M. Backbone Colorings and Generalized Mycielski Graphs. SIAM J. Discrete Math. 23 (2009), 1063-1070. https://doi.org/10.1137/080717596

    Togni O. Strong Chromatic Index of Products of Graphs. Discrete Math. and Theoretical Comp. Sci. 9 (2007), 47-56. https://doi.org/10.46298/dmtcs.414

    Thiru V.S., Balaji S. Strong Chromatic Indices of Certain Binary Operations on Graphs. Discrete Math., Algorithms Appl. 16 (2024), 2350073. https://doi.org/10.1142/S1793830923500738

  • Mathematics

    VERTEX DISTINGUISHING PROPER EDGE COLORINGS OF THE JOIN GRAPHS

    Tigran K. Petrosyan
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    Abstract

    A proper edge coloring of a graph $G$ is a mapping $f:E(G)\longrightarrow  \mathbb{Z}_{\geq 0}$ such that $f(e)\not=f(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. A proper edge coloring $f$ of a graph $G$ is called vertex distinguishing if for any different vertices $u,v \in V(G)$, $S(u,f) \ne S(v,f)$, where $S(v,f) = \{f(e) \ | \ e = wv \in E(G)\}$. The minimum number of colors required for a vertex distinguishing proper coloring of a graph $G$ is denoted by $\chi'_{vd}(G)$ and called vertex distinguishing chromatic index of $G$. In this paper we provide lower and upper bounds on the vertex distinguishing chromatic index of the join graphs.

    References

    West D.B. Introduction to Graph Theory. New Jersey, Prentice-Hall (2001).

    Burris A.C., Schelp R.H. Vertex-Distinguishing Proper Edge-colorings. J. Graph Theory 26 (1997), 73-82.

    Cerny J., Hornak M., Sotak R. Observability of a Graph. Math. Slovaca 46 (1996), 21-31.

    Burris A.C. Vertex-distinguishing Edge-colorings. Ph.D. Thesis. Memphis State University, Memphis, Tennessee (1993).

    Hornak M., Sotak R. Observability of Complete Multipartite Graphs with Equipotent Parts. Ars Combin. 41 (1995), 289-301.

    Balister P.N., Bollobas B., Schelp R.H. Vertex Distinguishing Coloring of Graphs with Δ(G)=2. Discrete Math. 252 (2002), 17-29.

    https://doi.org/10.1016/S0012-365X(01)00287-4

    Vizing V.G. On an Estimate of the Chromatic Class of a p-Graph. Diskrete Analiz. 3 (1964), 25-30 (in Russian).

    Zykov A.A. On Some Properties of Linear Complexes. Mat. Sbornik 24 (1949), 163-188 (in Russian).

    Baril J.-L., Kheddouci H., Togni O. Vertex Distinguishing Edge- and Total-colorings of Cartesian and Other Product Graphs. Ars Combin. 107 (2012), 109-127.

Physics

  • Physics

    STUDY OF  $\mathrm{PbWO}_4$  CRYSTAL FOR ePIC EmCal PROTOTYPE

    Argine S. Hakobyan, Diana G. Khurshudyan, Artak H. Mkrtchyan
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    Abstract

    The study presents the characterization of $\mathrm{PbWO}_4$ crystals intended for the prototype of the EmCal electromagnetic calorimeter of the ePIC detector, which is being constructed at the Electron-Ion Collider in the Brookhaven National Laboratory at the USA.  Measurements were performed on 20 crystals produced by "Crytur" company, each of which was then thoroughly examined under a microscope. Transversal transparency measurements were made at the center of the crystals, as well as at several fixed points equidistant from the center to the right and left sides to study uniformity. The average transparency of the crystals is 21.3%, 65.6%, and 71.7% for wavelengths of 360 nm, 440 nm, and 600 nm, respectively. The transmittance measurements repeated 10 times in the center of each crystal show that accuracy of our measurement is better than 10%. The light yield of $\mathrm{PbWO}_4$  was estimated to be an average 16 pe/MeV. The optical characteristics of Crytur crystals meet the requirements of the EIC electromagnetic calorimeter. After performing all the necessary measurements, 16 crystals in good condition were selected for the calorimeter prototype. A 4$\times$4 prototype of EmCal was designed, constructed and tested with cosmic muons.

    References

    Khalek R.A., et al. Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report. Nuclear Physics A 1026 (2021). https://doi.org/10.1016/j.nuclphysa.2021

    Adam J., et al. ATHENA Detector Proposal-a Totally Hermetic Electron Nucleus Apparatus Proposed for IP6 at the Electron-Ion Collider.

    Journal of Instrumentation 17 (2022), P10019. https://doi.org/10.1088/1748-0221/17/10/P10019

    Bylinkin A., et al. Detector Requirements and Simulation Results for the EIC Exclusive, Diffractive and Tagging Physics Program Using the ECCE Detector Concept. Nucl. Instrum. Meth. A (2023). https://doi.org/10.48550/arXiv.2208.14575

    Alarcon R., et al. CORE-a Compact Detector for the EIC. e-Print: 2209.00496 [physics.ins-det]

    Adzic P., Andelin D., Almeida N. The CMS Electromagnetic Calorimeter Group Radiation Hardness Qualification of $mathrm{PbWO}_4$ Scintillation Crystals for the CMS Electromagnetic Calorimeter.

    https://doi.org/10.1088/1748-0221/5/03/P03010

    Annenkov A.A., Korzhik M.V., Lecoq P. Nuclear Instruments and Methods in Physics Research A 490 (2002),

    30-50. Lead Tungstate Scintillation Material.

    Hakobyan A. Status of Electron Linear Accelerator LUE-75 of the A. Alikhanyan National Science Laboratory and Stability of Electron Beam Energy. J. Contemp. Phys. (Armenian Ac. Sci.) 56 (2021), 169. https://arar.sci.am/dlibra/publication/309999

    Dafinei I. Optical and Scintillation Properties of Lead Tungstate Crystals: A Statistical Approach. CERN Technical Report (2006), 019.

    Horn T., Berdnikov V.V., et al. Scintillating Crystals for the Neutral Particle Spectrometer in Hall C at JLab. Nucl. Instrum. and Methods A 956 (2020), 163375. https://doi.org/10.1016/j.nima.2019.163375