Vol. 59 No. 3 (268) (2025)

DOI: https://doi.org/10.46991/PYSUA.2025.59.3
Published: 2025-12-20

Mathematics

  • Mathematics

    ON THE CALCULATION OF THE COEFFICIENTS OF CUBIC SPLINES ON A SET OF EQUIDISTANT KNOTS

    Avetik H. Manukyan
    View PDF
    Abstract

    As is known, the coefficients of the interpolation cubic spline are found by solving a tridiagonal system of linear algebraic equations of a special type. To solve the system, a well-known numerical algorithm is usually used. In this paper, an alternative method for finding the coefficients of a natural cubic spline on a uniform set of knots is proposed. The method is based on the analytical inversion of the tridiagonal matrix, which made it possible to obtain closed-form expressions for the coefficients. This approach allows us both identify the analytical dependence of the spline coefficients on its values at the knots and obtain simple formulas for calculating these coefficients, by passing the solution of the system.

    References

    Kincaid D., Cheney W. Numerical Analysis. CA, Pacific Grove, Brooks/Cole (1991).

    Quarteroni A., Sacco R., Saleri F. Numerical Analysis. Springer (2007).

    Hakopian Yu.R., Manukyan A.H. Analytical Inversion of Tridiagonal Matrices. Mathematical Problems of Computer Science 58 (2022), 7-19.

  • Mathematics

    ON SUM EDGE-COLORINGS OF COMPLETE TRIPARTITE GRAPHS

    Hamlet V. Mikaelyan
    View PDF
    Abstract

    A proper edge-coloring of a graph is called a sum edge-coloring if it minimizes the total sum of colors on all the edges of the graph. The aforementioned minimal sum is called the edge-chromatic sum of the graph, and the minimal number of colors needed for a sum edge-coloring is called the edge-strength of the graph. In this paper, upper bounds on the values of the edge-chromatic sums of some complete tripartite graphs are given, while for some other complete tripartite graphs, the exact values of both parameters are obtained.

    References

    Supowit K.J. Finding a Maximum Planar Subset of a Set of Nets in a Channel. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 6 (1987), 93-94. https://doi.org/10.1109/TCAD.1987.1270250

    Kubicka E. The Chromatic Sum of a Graph. Western Michigan University (1989).

    Bar-Noy A., Bellare M., et al. On Chromatic Sums and Distributed Resource Allocation. Inform. and Comput. 140 (1998), 183-202. https://doi.org/10.1006/inco.1997.2677

    Salavatipour M.R. On Sum Coloring of Graphs. Discrete Appl. Math. 127 (2003), 477-488. https://doi.org/10.1016/S0166-218X(02)00249-4

    Giaro K., Kubale M. Edge-Chromatic Sum of Trees and Bounded Cyclicity Graphs. Inform. Process. Lett. 75 (2000), 65-69. https://doi.org/10.1016/S0020-0190(00)00072-7

    Petrosyan P.A., Kamalian R.R. On Sum Edge-Coloring of Regular, Bipartite and Split Graphs. Discrete Appl. Math. 165 (2014), 263-269. https://doi.org/10.1016/j.dam.2013.09.025

    Hajiabolhassan H., Mehrabadi M.L., Tusserkani R. Minimal Coloring and Strength of Graphs. Discrete Math. 215 (2000), 265-270. https://doi.org/10.1016/S0012-365X(99)00319-2

    Cardinal J., Ravelomanana V., Valencia-Pabon M. Chromatic Edge Strength of Some Multigraphs. Electron. Notes Discrete Math. 30 (2008), 39-44. https://doi.org/10.1016/j.endm.2008.01.008

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

    Hoffman D.G., Rodger C.A. The Chromatic Index of Complete Multipartite Graphs. J. Graph Theory 16 (1992), 159-163. https://api.semanticscholar.org/CorpusID:29985008

  • Mathematics

    ON DEFICIENCY OF COMPLETE 3-PARTITE AND 4-PARTITE GRAPHS

    Vahagn D. Tsirunyan
    View PDF
    Abstract

    A proper t-edge-coloring of a graph G is a mapping $\alpha: E(G)\rightarrow \{1,\ldots,t\}$ such that all colors are used, and $\alpha(e)\neq \alpha(e^{\prime})$ for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha $ is a proper edge-coloring of a graph G and $v\in V(G)$,  then the spectrum of a vertex v, denoted by $S\left(v,\alpha \right)$, is the set of all colors appearing on edges incident to v. The deficiency of $\alpha$ at vertex $v\in V(G)$,  denoted by $\mathrm{def}(v,\alpha)$, is the minimum number of integers that must be added to $S\left(v,\alpha \right)$ to form an interval, and the deficiency $\mathrm{def}\left(G,\alpha\right)$ of a proper edge-coloring $\alpha$ of G is defined as the sum $\displaystyle\sum_{v\in V(G)}\mathrm{def}(v,\alpha)$. The deficiency of a graph G, denoted by $\mathrm{def}(G)$, is defined as follows: $\mathrm{def}(G)=\min_{\alpha}\mathrm{def}\left(G,\alpha\right)$, where the minimum is taken over all possible proper edge-colorings of G. In 2019, Davtyan, Minasyan, and Petrosyan provided an upper bound on the deficiency of complete multipartite graphs. In this paper, we improve this bound for complete tripartite and some complete 4-partite graphs. We also confirm the conjecture that states the deficiency of a graph is bounded by the number of vertices of the graph for all tripartite graphs containing up to 10 vertices.

    References

    Asratian A.S., Kamalian R.R. Interval Colorings of Edges of a Multigraph. Appl. Math. 5 (1987), 25-34 (in Russian).

    Asratian A.S., Kamalian R.R. Investigation on Interval Edge-Colorings of Graphs. J. Combin. Theory Ser. B 62 (1994), 34-43. https://doi.org/10.1006/jctb.1994.1053

    Kamalian R.R. Interval Colorings of Complete Bipartite Graphs and Trees. Preprint. Comp. Cen. of Acad. Sci. of Armenian SSR. Yerevan (1989) (in Russian).

    Kamalian R.R. Interval Edge-Colorings of Graphs. Doctoral Thesis, Novosibirsk (1990).

    Petrosyan P.A. Interval Edge-Colorings of Complete Graphs and n-Dimensional Cubes. Discrete Math. 310 (2010), 1580-1587. https://doi.org/10.1016/j.disc.2010.02.001

    Petrosyan P.A., Khachatrian H.H., Tananyan H.G. Interval Edge-Colorings of Cartesian Products of Graphs. I. Discuss. Math. Graph Theory 33 (2013), 613-632. https://doi.org/10.48550/arXiv.1202.0023

    Sevast'janov S.V. Interval Colorability of the Edges of a Bipartite Graph. Metody Diskret. Analiza 50 (1990), 61-72 (in Russian).

    Asratia A.S., Denley T.M.J., Haggkvist R. Bipartite Graphs and their Applications. Cambridge University Press, Cambridge (1998).

    Kubale M. Graph Colorings. American Mathematical Society (2004).

    Giaro K., Kubale M., Malafiejski M. On the Deficiency of Bipartite Graphs. Discrete Appl. Math. 94 (1999), 193-203. https://doi.org/10.1016/S0166-218X(99)00021-9

    Petrosyan P.A., Sargsyan H.E. On Resistance of Graphs. Discrete Appl. Math. 159 (2011), 1889-1900. https://doi.org/10.1016/j.dam.2010.11.001

    Giaro K., Kubale M., Malafiejski M. Consecutive Colorings of the Edges of General Graphs. Discrete Math. 236 (2001), 131-143.

    Schwartz A. The Deficiency of a Regular Graph. Discrete Math. 306 (2006), 1947-1954. https://doi.org/10.1016/j.disc.2006.03.059

    Bouchard M., Hertz A., Desaulniers G. Lower bounds and a Tabu Search Algorithm for the Minimum Deficiency Problem. J. Comb. Optim. 17 (2009), 168-191. https://doi.org/10.1007/s10878-007-9106-0

    Borowiecka-Olszewska M., Drgas-Burchardt E., Haluszczak M. On the Structure and Deficiency of k-Trees with Bounded Degree. Discrete Appl. Math. 201 (2016), 24-37. https://doi.org/10.1016/j.dam.2015.08.008

    Davtyan A.R., Minasyan G.M., Petrosyan P.A. On the Deficiency of Complete Multipartite Graphs (2019). https://doi.org/10.48550/arXiv.1912.01546

    Petrosyan P.A., Khachatrian H.H., Mamikonyan T.K. On Interval Edge-Colorings of Bipartite Graphs. IEEE Computer Science and Information Technologies (CSIT) (2015), 71-76. https://doi.org/10.1109/CSITechnol.2015.7358253

    Tepanyan H.H., Petrosyan P.A. Interval Edge-Colorings of Composition of Graphs. Discrete Applied Mathematics 217 (2017), 368-374. https://doi.org/10.1016/j.dam.2016.09.022

    McKay B.D., Piperno A. Practical Graph Isomorphism, II. Journal of Symbolic Computation 60 (2014), 94-112. https://pallini.di.uniroma1.it

  • Mathematics

    COMPLEMENTS OF CLASSICAL AND DYNAMIC INEQUALITIES ANALYZED ON CALCULUS OF TIME SCALES

    M. Jibril Shahab Sahir
    View PDF
    Abstract

    In this research article, we present several generalizations of reverses of Callebaut's, Rogers-Hölder's and Cauchy-Schwarz's inequalities via reverses of Young's inequalities on time scales. Discrete, continuous, quantum versions of results are unified and extended on time scales.

    References

    Hilger S. Ein Mass kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. Thesis. Universität Würzburg (1988).

    Sahir M.J.S. Consonancy of Dynamic Inequalities Correlated on Time Scale Calculus. Tamkang Journal of Mathematics 51 (2020), 233-243. https://doi.org/10.5556/j.tkjm.51.2020.3145

    Sahir M.J.S. Reconciliation of Discrete and Continuous Versions of Some Dynamic Inequalities Synthesized on Time Scale Calculus. Communications in Mathematics 28 (2020), 277-287. https://doi.org/10.2478/cm-2020-0023

    Sahir M.J.S. Patterns of Time Scale Dynamic Inequalities Settled by Kantorovich's Ratio. Jordan Journal of Mathematics and Statistics (JJMS) 14 (2021), 397-410. https://doi.org/10.47013/14.3.1

    Sahir M.J.S. Coordination of Classical and Dynamic Inequalities Complying on Time Scales. Eur. J. Math. Anal. 3 (2023), Article 12. https://doi.org/10.28924/ada/ma.3.12

    Bohner M., Peterson A. Dynamic Equations on Time Scales. MA, Boston, Birkhäuser Boston, Inc. (2001).

    Bohner M., Peterson A. Advances in Dynamic Equations on Time Scales. MA, Boston, Birkhäuser Boston, Inc. (2003).

    Dragomir S.S. Some Results for Isotonic Functionals via an Inequality Due to Kittaneh and Manasrah. Fasciculi Mathematici 59 (2017), 29-42. https://doi.org/10.1515/fascmath-2017-0015

    Agarwal R.P., O'Regan D., Saker S.H. Dynamic Inequalities on Time Scales. Switzerland, Cham, Springer International Publishing (2014).

    Sheng Q., Fadag M., Henderson J., Davis J.M. An Exploration of Combined Dynamic Derivatives on Time Scales and Their Applications. Nonlinear Anal. Real World Appl. 7 (2006), 395-413. https://doi.org/10.1016/j.nonrwa.2005.03.008

    Kittaneh F., Manasrah Y. Improved Young and Heinz Inequalities for Matrices. J. Math. Anal. Appl. 361 (2010), 262-269. https://doi.org/10.1016/j.jmaa.2009.08.059

    Kittaneh F., Manasrah Y. Reverse Young and Heinz Inequalities for Matrices. Linear Multilinear Algebra 59 (2011), 1031-1037. https://api.semanticscholar.org/CorpusID:119892717

    Dragomir S.S. Some Results for Isotonic Functionals via an Inequality Due to Liao, Wu and Zhao. RGMIA Res. Rep. Coll. 18 (2015), 11. https://doi.org/10.1515/fascmath-2017-0015