2020
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Kato's chaos and Pchaos of a coupled lattice system given by Garcia Guirao and Lampart which is related with BelusovZhabotinskii reaction
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2
In this article, we further consider the above system. In particular, we give a sufficient condition under which the above system is Kato chaotic for $eta=0$ and a necessary condition for the above system to be Kato chaotic for $eta=0$. Moreover, it is deduced that for $eta=0$, if $Theta$ is Pchaotic then so is this system, where a continuous map $Theta$ from a compact metric space $Z$ to itself is said to be Pchaotic if it has the pseudoorbittracing property and the closure of the set of all periodic points for $Theta$ is the space $Z$. Also, an example and three open problems are presented.
1

1
9


Risong
Li
Guangdong Ocean University
Guangdong Ocean University
P. R. China
gdoulrs@163.com
Coupled map lattice
Kato's chaos
Pchaos
LiYorke's chaos
Tent map
[T. Y. Li and J. A. Yorke, Period three implies chaos, Am. Math. Mon. 82 (10) (1975) 985992.##L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics, Springer, Berlin, 1992.##R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/ Cumings, Menlo Park, CA, 1986.##J. R. Chazottes and B. Fern Andez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics (Berlin Heidelberg New York Springer), Vol. 671, 2005.##J. L. García Guirao and M. Lampart, Positive entropy of a coupled lattice system related with BelusovZhabotinskii reaction, J. Math. Chem. 48 (2010) 6671.##K. Kaneko, Globally coupled chaos violates law of large numbers, Phys. Rev. Lett. 65 (1990) 13911394.##J. L. García Guirao and M. Lampart, Chaos of a coupled lattice system related with BelusovZhabotinskii reaction, J. Math. Chem. 48 (2010) 159164.##M. Kohmoto and Y. Oono, Discrete model of chemical turbulence, Phys. Rev. Lett. 55 (1985) 29272931.##J. L. Hudson, M. Hart and D. Marinko, An experimental study of multiplex peak periodic and nonperiodic oscilations in the BelusovZhabotinskii reaction, J. Chem. Phys. 71 (1979) 16011606.##K. Hirakawa, Y. Oono and H. Yamakazi, Experimental study on chemical turbulence II, J. Phys. Soc. Jap. 46 (1979) 721728.##J. L. Hudson, K. R. Graziani, R. A. Schmitz, Experimental evidence of chaotic states in the BelusovZhabotinskii reaction, J. Chem. Phys. 67 (1977) 30403044.##D. Ruelle and F. Takens, On the natural of turbulence, Comm. Math. Phys. 20 (1971) 16792.##H. Kato, Everywhere chaotic homeomorphisms on manifields and kdimensional Menger manifolds, Topol. Appl. 72 (1996) 117.##R. Gu, Kato’s chaos in setvalued discrete systems, Chaos, Solitons & Fractals 31 (2007) 765771.##G. L. Forti, Various notions of chaos for discrete dynamical systems, A brief survey, Aequationes Math. 70 (2005) 113.##X. Wu and P. Zhu, On sensitive dependence of continuous interval mappings (In Chinese), J. Systems Sci. Math. Sci. 32 (2012) 215225.##T. Arai and N. Chinen, Pchaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topol. Appl. 154 (2007) 12541262. ##X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29 (2016) 19421972.##F. Balibrea, On problems of topological dynamics in nonautonomous discrete systems, Appl. Math. Nonlinear Sci. 1 (2) (2016) 391404.##]
On Topological Properties of the nStar Graph
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2
The nstar graph Sn is defined on the set of all n sequenses (u1,u2,...,un), ui ∈ {1, 2, ..., n}, ui ne uj and i ne j, where edges are of the form (u1,u2,...,un) ∼ (ui,u2,...,un), for some i ne 1. In this paper we will show that Sn is a vertex and edge transitive graph and discuss some topological properties of Sn.
1

11
16


Negur
Karamzadeh
Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran
Department of Mathematics, Faculty of Mathematical
Iran
n_shahni@sbu.ac.ir


Mohammad
Darafsheh
University of Tehran
University of Tehran
Iran
darafsheh@ut.ac.ir
Star graph
Vertex transitive graph
edge transitive graph
Wiener index
[S. B. Akers, D. Harel and B. Krishnamurthy, The star graph: An attractive alternative to the ncube, Proc. International Conference on Parallel Processing, St. Charles, Illinois, 1987, pp. 393400.##W. K. Chiang and R. J. Chen, The (n, k)star graph: A generalized star graph, Info. Proc. Lett. 56 (1995) 259264.##M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math. 110 (2010) 12251235.##I. Gutman, S. Klavžar and B. Mohr (eds), Fifty years of the Wiener index, MATCH Commun. Math. Comput. Chem. 35 (1997) 1259.##I. Gutman, Y. N. Yeh, S. L. Lee and J. C. Chen, Wiener numbers of dendrimers, MATCH Commun. Math. Comput. Chem. 30 (1994)103115.##K. Qiu and S. G. Akl, On some properties of the star graph, VLSI Design, 2 (4) (1995) 389396.##H. Shabani and A. R. Ashrafi, SymmetryModerated Wiener index, MATCH Commun. Math. Comput Chem. 76 (2016) 318.##H. Wiener, Structural determination of paraffin boiling points, J. Am. Chm. Soc. 69 (1947) 1720.##]
A New Explicit Singularly PStable FourStep Method for the Numerical Solution of Second Order IVPs
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2
In this paper, we introduce a new symmetric explicit fourstep method with variable coefficients for the numerical solution of secondorder linear periodic and oscillatory initial value problems of ordinary differential equations. For the first time in the literature, we generate an explicit method with the most important singularly Pstability property. The method is multiderivative and has algebraic order eight and infinite order of phaselag. The numerical results for some chemical (e.g. orbit problems of Stiefel and Bettis) as well as quantum chemistry problems (i.e. systems of coupled differential equations) indicated that the new method is superior, efficient, accurate and stable.
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17
31


Mohammad
Mehdizadeh Khalsaraei
Department of mathematics, University of Maragheh, Amirkabir Highway, P. O. Box. 5518183111
Department of mathematics, University of
Iran
muhammad.mehdizadeh@gmail.com


Ali
Shokri
Department of Mathematics, Faculty of Basic Science, University of Maragheh, Maragheh, Iran.
Department of Mathematics, Faculty of Basic
Iran
shokri2090@gmail.com
Explicit methods
Phaselag
Ordinary differential equations
Pstable
Symmetric multistep methods
[A. C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970) 378391.##J. R. Dormand and P. J. Prince, A family of embedded RungeKutta formulae, J. Comput. Appl. Math. 6 (1) (1980) 1926.##J. M. Franco and M. Palacios, Highorder Pstable multistep methods, J. Comput. Appl. Math. 30 (1) (1990) 110.##F. Hui and T. E. Simos, A new family of two stage symmetric twostep methods with vanished phaselag and its derivatives for the numerical integration of the Schrödinger equation, J. Math. Chem. 53 (10) (2015) 21912213.##J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189202.##Q. Li and X. Y. Wu, A twostep explicit Pstable method for solving secondorder initial value problems, Appl. Math. Comput. 138 (23) (2003) 435442.##Q. Li and X. Y. Wu, A twostep explicit Pstable method of high phaselag order for second order IVPs, Appl. Math. Comput. 151 (1) (2004) 1726.##Q. Li and X. Y. Wu, A twostep explicit Pstable method of high phaselag order for linear periodic IVPs, J. Comput. Appl. Math. 200 (1) (2007) 287296.##M. Mehdizadeh Khalsaraei, A. Shokri and M. Molayi, The new high approximation of stiff systems of first order IVPs arising from chemical reactions by kstep Lstable hybrid methods, Iranian J. Math. Chem. 10 (2) (2019) 181193.##M. Mehdizadeh Khalsaraei and A. Shokri, An explicit sixstep singularly Pstable Obrechkoff method for the numerical solution of secondorder oscillatory initial value problems, Numer. Algor. (2019), DOI:10.1007/s1107501900784w.##K. Mu and T. E. Simos, A RungeKutta type implicit high algebraic order twostep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation, J. Math. Chem. 53 (5) (2015) 12391256.##B. Neta, Pstable symmetric superimplicit methods for periodic initial value problems, Comput. Math. Appl. 50 (56) (2005) 701705.##G. D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, The Astro. J. 100 (5) (1990) 16941700.##H. Ramos, Development of a new RungeKutta method and its economical implementation, Comput. Math. Methods 1 (2) (2019) e1016.##A. D. Rapits, Exponentiallyfitted solutions of the eigenvalues Schrödinger equation with automatic error control, Comput. Phys. Commun. 28 (1983) 427431.##A. D. Rapits and A. C. Allison, Exponentialfitting methods for the numerical solution of the Schrödinger equation, J. Comput. Phys. Commun. 14 (1978) 15.##A. D. Raptis and J. R. Cash, A variable step method for the numerical integration of the onedimensional Schrödinger equation, Comput. Phys.Commun. 36 (2) (1985) 113119.##A. Shokri, A new eightorder symmetric twostep multiderivative method for the numerical solution of secondorder IVPs with oscillation solutions, Numer. Algor. 77 (1) (2018) 95109.##A. Shokri, An explicit trigonometrically fitted tenstep method with phaselag of order infinity for the numerical solution of the radial Schrödinger equation, Appl. Comput. Math. 14 (1) (2015) 6374.##A. Shokri, The symmetric twostep Pstable nonlinear predictorcorrector methods for the numerical solution of second order initial value problems, Bull. Iranian Math. Soc. 41 (2015) 191205.##A. Shokri, M. Mehdizadeh Khalsaraei, M. Tahmourasi and R. GarciaRubio, A new family of threestage twostep Pstable multiderivative methods with vanished phaselag some of its derivatives for the numerical solution of radial Schrödinger equation and IVPs with oscillating solutions, Numer. Algor. 80 (2) (2018) 557593.##A. Shokri, M.Y. Rahimi Ardabili, S. Shahmorad and G. Hojjati, A new twostep Pstable hybrid Obrechkoff method for the numerical integration of secondorder IVPs., J. Comput. Appl. Math. 235 (2011) 17061712.##A. Shokri and H. Saadat, High phaselag order trigonometrically fitted twostep Obrechkoff methods for the numerical solution of periodic initial value problems, Numer. Algor. 68 (2015) 337354.##A. Shokri and M. Tahmourasi, A new twostep Obrechkoff method with vanished phaselag and some of its derivatives for the numerical solution of radial Schrödinger equation and related IVPs with oscillating solutions, Iranian J. Math. Chem. 8 (2) (2017) 137159.##A. Shokri, J. VigoAguiar, M. Mehdizadeh Khalsaraei and R. GarciaRubio, A new class of twostep Pstable TFPL methods for the numerical solution second order IVPs with oscillating solutions, J. Comput. Appl. Math. 354 (2019) 551561.##A. Shokri, J. VigoAguiar, M. Mehdizadeh Khalsaraei and R. GarciaRubio, A new fourstep Pstable Obrechkoff method with vanished phaselag and some of its derivatives for the numerical solution of Schrödinger equation, J. Comput. Appl. Math. 354 (2019) 569586.##A. Shokri, J. VigoAguiar, M. Mehdizadeh Khalsaraei and R. GarciaRubio, A new implicit sixstep Pstable method for the numerical solution of Schrödinger equation, Int. J. Comput. Math. (2019), DOI: 10.1080/00207160.2019.1588257.##S. Stavroyiannis and T. E. Simos, A nonlinear explicit twostep algebraic order method of order infinity for linear periodic initial value problems, Comput. Phys. Commun. 181 (8) (2010) 13621368.##S. Stavroyiannis and T. E. Simos, Optimization as a function of the phaselag order of nonlinear explicit twostep Pstable method for linear periodic IVPs, Appl. Numer. Math. 59 (10) (2009) 24672474.##T. E. Simos, Exponentially fitted RungeKutta methods for the numerical solution of the Schrödinger equation and related problems, Comput. Mater. Sci. 18 (34) (2000) 315332.##T. E. Simos and J. VigoAguiar, An exponential fitted high order method for longtime integration of periodic initialvalue problems, Comput. Phys. Commun. 140 (3) (2001) 358365.##T. E. Simos and P. S. Williams, A finitedifference method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math., 79 (2) (1997) 189205.##E. Steifel and D. G. Bettis, Stabilization of Covell’s methods, Numer. Math. 13 (1969) 154175.##J. VigoAguiar and H. Ramos, Variable stepsize implementation of multistep methods for , J. Comput. Appl. Math. 192 (2006) 114131.##X. Xi and T. E. Simos, A new fourstages twelfth algebraic order twostep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation, MATCH Commun. Math. Comput. Chem. 77 (2) (2017) 333392.##Z. Zhou and T. E. Simos, A new two stage symmetric twostep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation, J. Math. Chem. 54 (2) (2016) 442465.##]
Some New Results on Mostar Index of Graphs
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2
A general bond additive index (GBA) can be defined as , where α(e) is edge contributions. The Mostar index is a new topological index whose edge contributions are α(e) =  nu  nv in which nu is the number of vertices of lying closer to vertex u than to vertex v and nv can be defined similarly. In this paper, we propose some new results on the Mostar index based on the vertexorbits under the action of automorphism group. In addition, we detrmined the structures of graphs with Mostar index equal 1. Finally, compute the Mostar index of a family of nanocone graphs.
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33
42


Modjtaba
Ghorbani
Department of mathematics, Shahid Rajaee Teacher Training University
Department of mathematics, Shahid Rajaee
Iran
mghorbani@sru.ac.ir


Shaghayegh
Rahmani
Department of Mathematics, SRTT University
Department of Mathematics, SRTT University
Iran
s.rahmani@sru.ac.ir


Mohammad
Eslampoor
Department of Mathematics, Srtt University
Department of Mathematics, Srtt University
Iran
ag.paper@gmail.com
Molecular graphs
topological index
Automorphism group
[J. Bok, B. Furtula, N. Jedlickova and R. Skrekovski, On extremal graphs of weighted Szeged index, MATCH Commun. Math. Comput. Chem. 82 (2019) 93−109.## J. Devillers and A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Amsterdam, Netherlands, Gordon and Breach, 2000.##M. V. Diudea and I. Gutman, Wienertype topological indices, Croat. Chem. Acta 71 (1) (1998) 21−51.##D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263−267.##A. A. Dobrynin, On the Wiener index of certain families of fibonacenes, MATCH Commun. Math. Comput. Chem. 70 (2013) 565−574.##A. A. Dobrynin, The Szeged and Wiener indices of line graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 743−756##A. A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math. Beograd. 56 (1994) 18−22.##T. Došlić, I. Martinjak, R. Škrekovski, S. Tipurić Spužević and I. Zubac, Mostar index, J. Math. Chem. (2018), DOI:10.1007/s109100180928z.##M. Ghorbani, X. Li, H. R. Maimani, Y. Mao, S. Rahmani and M. RajabiParsa, Steiner (Revised) Szeged index of graphs, MATCH Commun. Math. Comput. Chem. 82 (2019) 733−742.##M. Ghorbani; M. Songhori, Some topological indices of nanostar dendrimers, Iranian J. Math. Chem. 1 (2010) 57−65.##M. Ghorbani and S. Rahmani, A note on Mostar index of a class of fullerenes, Int. J. of Chem. Model. 9 (2017) 245−256.##I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York 27 (1994) 9−15.##I. Gutman and A. A. Dobrynin, The Szeged indexa success story, Graph Theory Notes New York 34 (1998) 37−44.##I. Gutman, P. V. Khadikar, P. V. Rajput and S. Karmarkar, The Szeged index of polyacenes, J. Serb. Chem. Soc. 60 (1995) 759−764.##I. Gutman, S. Klavžar and B. Mohar (Eds), Fifty years of the Wiener index, MATCH Commun. Math. Comput. Chem. 35 (1997) 12−59.##I. Gutman, Y. N. Yeh, S. L. Lee and Y. L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993) 651−661.##S. Klavžar and I. Gutman, A theorem on Wienertype invariants for isometric subgraphs of hypercubes, Appl. Math. Lett. 19 (2006) 1129−1133.##S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (1996) 45−49.##X. Li and M. Zhang, A note on the computation of revised (edge)Szeged index in terms of canonical isometric embedding, MATCH Commun. Math. Comput. Chem. 81 (2019) 149−162.##G. Rücker and C. Rücker, On topological indices, boiling points and cycloalkanes, J. Chem. Inf. Comput. Sci. 39 (1999) 788−802.##B. E. Sagan, Y.N. Yeh and P. Zhang, The Wiener polynomial of a graph, Int. J. Quantum. Chem. 60 (1996) 959−969.##R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley−VCH, Weinheim, 2000.##H. J. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17−20.##]
A Robust Spectrophotometric Method using Least Squares Support Vector Machine for Simultaneous Determination of Anti−Diabetic Drugs and Comparison with the Chromatographic Method
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2
In the present paper, the simultaneous spectrophotometric estimation of Metformin (MET) and Pioglitazone (PIO) in an antidiabetic drug called Actoplus MET based on least squares support vector machine (LSSVM) was proposed. The optimum gamma (γ) and sigma (σ) parameters were found to be 825 and 90 with the root mean square error (RMSE) of 0.1343for MET, as well as 1000 and 350 with RMSE=0.4120 for PIO. Also, the mean recovery values of MET and PIO were 99.81% and 100.19%, respectively. Ultimately, the real sample was analyzed by HighPerformance Liquid Chromatography (HPLC) reference method and the proposed procedure. Then, oneway analysis of variance (ANOVA) test at the 95 % confidence level was performed on achieved results from HPLC and LSSVM methods. The statistical data of these methods showed that there were no significant differences between them.
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43
55


Valeh
Arabzadeh
Department of Chemistry, North Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Chemistry, North Tehran Branch,
Iran
valeh_ar@yahoo.com


Mahmoud
Sohrabi
Department of Chemistry, North Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Chemistry, North Tehran Branch,
Iran
sohrabi.m46@yahoo.com


Nasser
Goudarzi
Faculty of chemistry, Shahrood University of Technology,
Shahrood, Iran
Faculty of chemistry, Shahrood University
Iran
goudarzi@shahroodut.ac.ir


Mehran
Davallo
Department of Chemistry, North Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Chemistry, North Tehran Branch,
Iran
mehrandavallo@yahoo.com
spectrophotometric
Least squares support vector machine
Metformin
Pioglitazone
Highperformance liquid chromatography
[S. A. Kumar, M. Debnath, J. V. L. N. S. Rao and D. G. Sankar, Simultaneous estimation of metformin, pioglitazone and glimepiride in bulk samples and in tablet dosage forms by using RPHPLC in an isocratic mode, J. Chem. Pharm. Res. 7 (1) (2015) 941951.##A. M. I. Mohamed, F. A. F. Mohamed, S. Ahmed and Y. A. S. Mohamed, An efficient hydrophilic interaction liquid chromatographic method for the simultaneous determination of metformin and pioglitazone using highpurity silica column, J. Chromatogr. B. 997 (2015) 16–22.## M. M. Sebaiy, S. M. ElAdl, M. M. Baraka and A. A. Hassan, Rapid RPHPLC method for simultaneous estimation of metformin, pioglitazone, and glimepiride in human plasma, Acta Chromatographica. 32 (1) (2019) 16.##S. Havele and S. Dhaneshwar, Development and validation of a HPLC method for the determination of metformin hydrochloride, gliclazide and pioglitazone hydrochloride in multicomponent formulation, WebmedCentral. Pharm. Sci. 1 (10) (2010) WMC0010.##A. D. Mali, S. Mali, A. Tamboli and R. Bathe, Simultaneous UV spectrophotometric methods for estimation of metformin HCl and glimepiride in bulk and tablet dosage form, Int. J. Adv. Pharm. 4 (6) (2015) 117124.##M. KawaguchiSuzuki, F. Bril, P. P. Sanchez, K. Cusi and R. F. Frye, A validated liquid chromatography tandem mass spectrometry method for simultaneous determination of pioglitazone, hydroxyl pioglitazone, and keto pioglitazone in human plasma and its application to a clinical study, J. Chromatogr. B. 969 (2014) 219–223.##P. K. Chaturvedi and R. Sharma, Simultaneous spectrophotometric estimation and validation of three component tablet formulation containing pioglitazone hydrochloride, metformin hydrochloride and glibenclamide, Anal. Lett. 41 (12) (2008) 2133–2142.##K. S. Lakshmi, T. Rajesh and S. Sharma, Simultaneous determination of metformin and pioglitazone by reversed phase HPLC in pharmaceutical dosage forms, Int. J. Pharm. Pharm. Sci. 1 (2)(2009) 162166.##G. S. Talele, D. D. Anghore and P. K. Porwal, Liquid chromatographic method for simultaneous estimation of metformin HCl, pioglitazone HCl and glibenclamide in rat plasma, Pharm Aspire. 10 (2018) 4147.##R. Peraman, K. K. Peruru, P. R. Yiragam and C. S. Gowra, Stability indicating RPHPLC method for the simultaneous determination of atorvastatin calcium, metformin hydrochloride, and glimepiride in bulk and combined tablet dosage form, Malays. J. Pharm. Sci. 12 (2014) 33–46.##G. S. Sandhu, S. S. Hallan and B. Kaur, Development of RPHPLC method for simultaneous estimation of glimepiride, pioglitazone hydrochloride and metformin hydrochloride in a combined tablet dosage form, World J. Pharm. Pharm. Sci. 5 (2016) 12781285.##S. A. Mulchand and B. R. Balkrishna, Novel RPHPLC method development and validation for simultaneous estimation of metformin, voglibose and pioglitazone in bulk and triple fixed drug combinations pharmaceutical dosage form, J. Drug. Deliv. Ther. 9 (2019) 3037.##M. R. Rezk, S. M. Riad, G. Y. Mahmoud and A. A. Aleem, Simultaneous determination of pioglitazone and glimepiride in their pharmaceutical formulations, Der Pharma Chem. 3 (5) (2011) 176184.##A. Onal, Spectrophotometric and HPLC determinations of antidiabetic drugs, rosiglitazone maleate and metformin hydrochloride, in pure form and in pharmaceutical preparations, Eur. J. Med. Chem. 44 (12) (2009) 4998–5005.##A. Khorshid, N. S. Abdelhamid, E. A. Abdelaleem and M. M. Amin, Simultaneous Determination of metformin and pioglitazone in presence of metformin impurity by different spectrophotometric and TLC – densitometric methods, SOJ. Pharm. Pharm. Sci. 5 (3) (2018) 18.##J. V. Susheel, D. Paul and T. K. Ravi, Development and validation of highperformance thinlayer chromatography method for the simultaneous densitometric determination of metformin and rosiglitazone in tablets, Austin J. Anal. Pharm. Chem. 3 (3) (2016) 10711074.##M. A. Hegazy, M. R. ElGhobashy, A. M. Yehia and A. A. Mostafa, Simultaneous determination of metformin hydrochloride and pioglitazone hydrochloride in binary mixture and in their ternary mixture with pioglitazone acid degradate using spectrophotometric and chemometric methods, Drug Test. Analysis. 1 (7) (2009) 339–349.##H. M. Lotfy, D. Mohamed and S. Mowaka, A comparative study of smart spectrophotometric methods for simultaneous determination of sitagliptin phosphate and metformin hydrochloride in their binary mixture, Spectrochim. Acta A Mol. Biomol. Spectrosc. 149 (2015) 441451.##J. A. K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural Process. Lett. 9 (3) (1999) 293–300.##Sh. Mofavvaz, M. R. Sohrabi and A. NezamzadehEjhieh, New model for prediction binary mixture of antihistamine decongestant using artificial neural networks and least squares support vector machineby spectrophotometry method, Spectrochim. Acta A Mol. Biomol. Spectrosc. 182 (2017) 105–115.##A. Baghban, M. Bahadori, A. S. Lemraski and A. Bahadori, Prediction of solubility of ammonia in liquid electrolytes using least square support vector machines, Ain Shams Eng. J. 9 (4)(2018) 1303–1312.##H. Han, X. Cui, Y. Fan and H. Qing, Least squares support vector machine (LSSVM)based chiller fault diagnosis using fault indicative features, Appl. Therm. Eng. 154 (2019) 540547. ##M. R. Sohrabi and G. Darabi, The application of continuous wavelet transform and least squares support vector machine for the simultaneous quantitative spectrophotometric determination of Myricetin, Kaempferol and Quercetin as flavonoids in pharmaceutical plants, Spectrochim. Acta A Mol. Biomol. Spectrosc. 152 (2016) 443–452.##J. Guan, J. Zurada and A. S. Levitan, An adaptive neurofuzzy inference system based approach to real estate property assessment, J. Real Estate Res. 30 (4) (2008) 395421.##J. N. Miller and J. C. Miller, Statistics and Chemometrics for Analytical Chemistry, 6th Ed. Pearson Education Limited, Essex, England, 2010.##]
On the Laplacian Szeged spectrum of paths
2
2
We present explicit formulas for the Laplacian Szeged eigenvalues of paths, grids, $C_4$nanotubes and of Cartesian products of paths with some other simple graphs. A number of open problems is listed.
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63


Tomislav
Doslic
Faculty of Civil Engineering, University of Zagreb
Faculty of Civil Engineering, University
Croatia
doslic@grad.hr
Path graph
grid graph
Cartesian product
[G. E. Andrews, W. Gawronski and L. L. Littlejohn, The LegendreStirling numbers, Discrete Math. 311 (2011) 1255–1272. ##N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974. ##M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Phys. A: Math. Theor. 40 (2007) 3815–3829. ##D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs, Academic Press, New York, 1980. ##G. H. FathTabar, T. Došlić and A. R. Ashrafi, On the Szeged and the Laplacian Szeged spectrum of a graph, Linear Algebra Appl. 433 (2010) 662–671. ##C. M. da Fonseca and E. Kiliç, A new type of Sylvester–Kac matrix and its spectrum, Linear Multilinear Algebra, to appear. ##C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993. ##I. Gutman, A formula for the Wiener number of trees and its extension to the graphs containing cycles, Graph Theory Notes of New York XVII (1994) 9–15. ##I. Gutman, A. A. Dobrynin, The Szeged indexa success story, Graph Theory Notes New York 34 (1998) 37–44. ##J. Schwinger, On Angular Momentum, Unpublished Report, Harvard University, Nuclear Development Associates, Inc., United States Department of Energy (through predecessor agency the Atomic Energy Commission), Report Number NYO3071 (January 26, 1952). ## N. J. A. Sloane, (ed.), The OnLine Encyclopedia of Integer Sequences, published electronically at http://oeis.org. ##D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. ##H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. ##]