ijpam.uniud.itijpam.uniud.it/online_issue/ijpam_no-44-2020.pdf · n° 44 – july 2020 italian...

N° 44 – July 2020

Italian Journal of Pure and

Applied Mathematics

ISSN 2239-0227

EDITOR-IN-CHIEF

Piergiulio Corsini

Editorial Board

Saeid Abbasbandy

Praveen Agarwal

Bayram Ali Ersoy

Reza Ameri

Luisa Arlotti

Alireza Seyed Ashrafi

Krassimir Atanassov

Vadim Azhmyakov

Malvina Baica

Federico Bartolozzi

Rajabali Borzooei

Carlo Cecchini

Gui-Yun Chen

Domenico Nico Chillemi

Stephen Comer

Irina Cristea

Mohammad Reza Darafsheh

Bal Kishan Dass

Bijan Davvaz

Mario De Salvo

Alberto Felice De Toni

Franco Eugeni

Mostafa Eslami

Giovanni Falcone

Yuming Feng

Antonino Giambruno

Furio Honsell

Luca Iseppi

James Jantosciak

Tomas Kepka

David Kinderlehrer

Sunil Kumar

Andrzej Lasota

Violeta Leoreanu-Fotea

Maria Antonietta Lepellere

Mario Marchi

Donatella Marini

Angelo Marzollo

Antonio Maturo

Fabrizio Maturo

Sarka Hozkova-Mayerova

Vishnu Narayan Mishra

M. Reza Moghadam

Syed Tauseef Mohyud-Din

Petr Nemec

Vasile Oproiu

Livio C. Piccinini

Goffredo Pieroni

Flavio Pressacco

Sanja Jancic Rasovic

Vito Roberto

Gaetano Russo

Paolo Salmon

Maria Scafati Tallini

Kar Ping Shum

Alessandro Silva

Florentin Smarandache

Sergio Spagnolo

Stefanos Spartalis

Hari M. Srivastava

Yves Sureau

Carlo Tasso

Ioan Tofan

Aldo Ventre

Thomas Vougiouklis

Hans Weber

Shanhe Wu

Xiao-Jun Yang

Yunqiang Yin

Mohammad Mehdi Zahedi

Fabio Zanolin

Paolo Zellini

Jianming Zhan

FORUM

EDITOR-IN-CHIEF

Piergiulio Corsini

Department of Civil Engineering and Architecture

Via delle Scienze 206 - 33100 Udine, Italy [email protected]

VICE-CHIEFS

Irina Cristea

Violeta Leoreanu


MANAGING BOARD

Domenico Chillemi, CHIEF Piergiulio Corsini

Irina Cristea Alberto Felice De Toni

Furio Honsell

Violeta Leoreanu Maria Antonietta Lepellere

Livio Piccinini

Flavio Pressacco

Luminita Teodorescu Norma Zamparo

EDITORIAL BOARD

Saeid Abbasbandy Dept. of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran [email protected] Praveen Agarwal Department of Mathematics, Anand International College of Engineering Jaipur-303012, India [email protected]

Bayram Ali Ersoy Department of Mathematics, Yildiz Technical University

34349 Beşiktaş, Istanbul, Turkey [email protected] Reza Ameri Department of Mathematics University of Tehran, Tehran, Iran [email protected]

Luisa Arlotti Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Alireza Seyed Ashrafi Department of Pure Mathematics University of Kashan, Kāshān, Isfahan, Iran

[email protected] Krassimir Atanassov Centre of Biomedical Engineering, Bulgarian Academy of Science BL 105 Acad. G. Bontchev Str. 1113 Sofia, Bulgaria [email protected] Vadim Azhmyakov

Department of Basic Sciences, Universidad de Medellin, Medellin, Republic of Colombia [email protected] Malvina Baica University of Wisconsin-Whitewater Dept. of Mathematical and Computer Sciences Whitewater, W.I. 53190, U.S.A.

[email protected] Federico Bartolozzi Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected] Rajabali Borzooei

Department of Mathematics Shahid Beheshti University, Tehran, Iran [email protected] Carlo Cecchini Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Gui-Yun Chen School of Mathematics and Statistics, Southwest University, 400715, Chongqing, China [email protected] Domenico (Nico) Chillemi Executive IT Specialist, IBM z System Software IBM Italy SpA

Via Sciangai 53 – 00144 Roma, Italy [email protected] Stephen Comer Department of Mathematics and Computer Science The Citadel, Charleston S. C. 29409, USA [email protected]

Irina Cristea CSIT, Centre for Systems and Information Technologies University of Nova Gorica Vipavska 13, Rožna Dolina, SI-5000 Nova Gorica, Slovenia [email protected]

Mohammad Reza Darafsheh School of Mathematics, College of Science

University of Tehran, Tehran, Iran [email protected] Bal Kishan Dass Department of Mathematics University of Delhi, Delhi - 110007, India [email protected]

Bijan Davvaz Department of Mathematics, Yazd University, Yazd, Iran [email protected] Mario De Salvo Dipartimento di Matematica e Informatica Viale Ferdinando Stagno d'Alcontres 31, Contrada Papardo 98166 Messina

[email protected] Alberto Felice De Toni Udine University, Rector Via Palladio 8 - 33100 Udine, Italy [email protected]

Franco Eugeni Dipartimento di Metodi Quantitativi per l'Economia del Territorio Università di Teramo, Italy [email protected]

Mostafa Eslami Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran [email protected] Giovanni Falcone Dipartimento di Metodi e Modelli Matematici

viale delle Scienze Ed. 8 90128 Palermo, Italy [email protected] Yuming Feng College of Math. and Comp. Science, Chongqing Three-Gorges University, Wanzhou, Chongqing, 404000, P.R.China [email protected]

Antonino Giambruno Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected] Furio Honsell Dipartimento di Matematica e Informatica

Via delle Scienze 206 - 33100 Udine, Italy [email protected] Luca Iseppi Department of Civil Engineering and Architecture, section of Economics and Landscape Via delle Scienze 206 - 33100 Udine, Italy [email protected]

James Jantosciak Department of Mathematics, Brooklyn College (CUNY) Brooklyn, New York 11210, USA [email protected] Tomas Kepka MFF-UK Sokolovská 83

18600 Praha 8,Czech Republic [email protected] David Kinderlehrer Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA15213-3890, USA [email protected]

Sunil Kumar Department of Mathematics, National Institute of Technology Jamshedpur, 831014, Jharkhand, India [email protected] Andrzej Lasota Silesian University, Institute of Mathematics Bankova 14

40-007 Katowice, Poland [email protected] Violeta Leoreanu-Fotea Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected]

Maria Antonietta Lepellere Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Mario Marchi Università Cattolica del Sacro Cuore via Trieste 17, 25121 Brescia, Italy

[email protected] Donatella Marini Dipartimento di Matematica Via Ferrata 1- 27100 Pavia, Italy [email protected] Angelo Marzollo

Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Antonio Maturo University of Chieti-Pescara, Department of Social Sciences, Via dei Vestini, 31 66013 Chieti, Italy

[email protected] Fabrizio Maturo University of Chieti-Pescara, Department of Management and Business Administration, Viale Pindaro, 44 65127 Pescara, Italy [email protected]

Sarka Hoskova-Mayerova Department of Mathematics and Physics University of Defence Kounicova 65, 662 10 Brno, Czech Republic [email protected] Vishnu Narayan Mishra Applied Mathematics and Humanities Department

Sardar Vallabhbhai National Institute of Technology 395 007, Surat, Gujarat, India [email protected] M. Reza Moghadam Faculty of Mathematical Science, Ferdowsi University of Mashhadh P.O.Box 1159 - 91775 Mashhad, Iran

[email protected] Syed Tauseef Mohyud-Din Faculty of Sciences, HITEC University Taxila Cantt Pakistan [email protected] Petr Nemec Czech University of Life Sciences, Kamycka’ 129 16521 Praha 6, Czech Republic [email protected] Vasile Oproiu Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected] Livio C. Piccinini Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Goffredo Pieroni Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy

[email protected]

Flavio Pressacco Dept. of Economy and Statistics Via Tomadini 30 33100, Udine, Italy [email protected] Sanja Jancic Rasovic Department of Mathematics Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjska 2 – 81000 Podgorica, Montenegro [email protected] Vito Roberto

Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Gaetano Russo Department of Civil Engineering and Architecture Via delle Scienze 206 33100 Udine, Italy

[email protected] Paolo Salmon Dipartimento di Matematica, Università di Bologna Piazza di Porta S. Donato 5 40126 Bologna, Italy [email protected]

Maria Scafati Tallini Dipartimento di Matematica "Guido Castelnuovo" Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected] Kar Ping Shum Faculty of Science The Chinese University of Hong Kong

Hong Kong, China (SAR) [email protected] Alessandro Silva Dipartimento di Matematica "Guido Castelnuovo", Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected]

Florentin Smarandache Department of Mathematics, University of New Mexico Gallup, NM 87301, USA [email protected] Sergio Spagnolo Scuola Normale Superiore Piazza dei Cavalieri 7 - 56100 Pisa, Italy

[email protected] Stefanos Spartalis Department of Production Engineering and Management, School of Engineering, Democritus University of Thrace V.Sofias 12, Prokat, Bdg A1, Office 308 67100 Xanthi, Greece [email protected] Hari M. Srivastava Department of Mathematics and Statistics University of Victoria, Victoria, British Columbia V8W3P4, Canada [email protected] Yves Sureau 27, rue d'Aubiere

63170 Perignat, Les Sarlieve - France [email protected] Carlo Tasso Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Ioan Tofan Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected] Aldo Ventre Seconda Università di Napoli, Fac. Architettura, Dip. Cultura del Progetto

Via San Lorenzo s/n 81031 Aversa (NA), Italy [email protected] Thomas Vougiouklis Democritus University of Thrace, School of Education, 681 00 Alexandroupolis. Greece

[email protected] Hans Weber Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Shanhe Wu Department of Mathematics, Longyan University,

Longyan, Fujian, 364012, China [email protected] Xiao-Jun Yang Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, China [email protected]

Yunqiang Yin School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, P.R. China [email protected]

Mohammad Mehdi Zahedi

Department of Mathematics, Faculty of Science Shahid Bahonar, University of Kerman Kerman, Iran [email protected] Fabio Zanolin Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy

[email protected] Paolo Zellini Dipartimento di Matematica, Università degli Studi Tor Vergata via Orazio Raimondo (loc. La Romanina) 00173 Roma, Italy [email protected]

Jianming Zhan Department of Mathematics, Hubei Institute for Nationalities Enshi, Hubei Province,445000, China [email protected]

mailto:[email protected]:[email protected]:[email protected]:[email protected]

i ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

In memoriam of Professor Giuseppe Manuppella

The “Italian Journal of Pure and Applied Mathematics“ cannot more take advantage of the

precious collaboration of prof. Giuseppe Manuppella, who has passed away, for

Mathematics event organization.

The members of Editorial Board express their deep sorrow for this loss.

The Managing Board Chief regrets the loss of Prof. Giuseppe Manuppella. He has been a

great man of science and a very dear friend.

All they who knew him will remember always his scientific value and his wonderful

human qualities.

Piergiulio Corsini

Domenico Chillemi

ii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

iii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

Italian Journal of Pure and Applied Mathematics ISSN 2239-0227

Web Site

http://ijpam.uniud.it/journal/home.html

Twitter @ijpamitaly

https://twitter.com/ijpamitaly

EDITOR-IN-CHIEF

Piergiulio Corsini

Department of Civil Engineering and Architecture

Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Vice-CHIEFS Irina Cristea

Violeta Leoreanu-Fotea Maria Antonietta Lepellere

Managing Board

Domenico Chillemi, CHIEF Piergiulio Corsini

Irina Cristea

Alberto Felice De Toni

Furio Honsell

Violeta Leoreanu-Fotea


Livio Piccinini Flavio Pressacco

Luminita Teodorescu

Norma Zamparo

Editorial Board

Saeid Abbasbandy

Praveen Agarwal Bayram Ali Ersoy

Reza Ameri

Luisa Arlotti Alireza Seyed Ashrafi

Krassimir Atanassov

Vadim Azhmyakov Malvina Baica

Federico Bartolozzi

Rajabali Borzooei Carlo Cecchini

Gui-Yun Chen

Domenico Nico Chillemi Stephen Comer

Irina Cristea

Mohammad Reza Darafsheh Bal Kishan Dass

Bijan Davvaz

Mario De Salvo Alberto Felice De Toni

Franco Eugeni

Mostafa Eslami Giovanni Falcone

Yuming Feng

Antonino Giambruno

Furio Honsell Luca Iseppi

James Jantosciak

Tomas Kepka David Kinderlehrer

Sunil Kumar

Andrzej Lasota Violeta Leoreanu-Fotea


Mario Marchi Donatella Marini

Angelo Marzollo

Antonio Maturo Fabrizio Maturo

Sarka Hozkova-Mayerova

Vishnu Narayan Mishra M. Reza Moghadam

Syed Tauseef Mohyud-Din

Petr Nemec Vasile Oproiu

Livio C. Piccinini

Goffredo Pieroni Flavio Pressacco

Sanja Jancic Rasovic

Vito Roberto

Gaetano Russo Paolo Salmon

Maria Scafati Tallini

Kar Ping Shum Alessandro Silva

Florentin Smarandache

Sergio Spagnolo Stefanos Spartalis

Hari M. Srivastava

Yves Sureau Carlo Tasso

Ioan Tofan

Aldo Ventre Thomas Vougiouklis

Hans Weber

Shanhe Wu Xiao-Jun Yang

Yunqiang Yin

Mohammad Mehdi Zahedi Fabio Zanolin

Paolo Zellini

Jianming Zhan

Forum Editrice Universitaria Udinese Srl

Via Larga 38 - 33100 Udine

Tel: +39-0432-26001, Fax: +39-0432-296756 [email protected]

http://ijpam.uniud.it/journal/home.htmlhttp://ijpam.uniud.it/journal/home.htmlmailto:[email protected]:[email protected]

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44–2020 iv

Table of contents

MD. Shakeel, Sharief Basha, Raja DasTwo-level secret sharing schemes based on reverse super edge magic labelings . . . . . . . . . . . . . 1–6

Ahmad Al-RhayyelEdge maximal W7-free graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7–21

S. Sara, M. AslamOn Lie ideals of inverse semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–29

Sh. Al-Sharif, A. MalkawiModification of conformable fractional derivative with classical properties . . . . . . . . . . . . . . 30–39

Baravan A. Asaad, Nazihah Ahmad, Zurni OmarProperties of γ-PS-R0 and γ-PS-R1 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40–56

Rashmi Bhardwaj, Saureesh DasSynchronization of two three-species food chain system

with Beddington-DeAngelis functional response using active controllersbased on the Lyapunov function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57–77

M. DavarzaniThe complexity of the graph access structures on seven participants . . . . . . . . . . . . . . . . . . . . 78–86

Guangwang Su, Taixiang Sun, Bin QinThe convergence of the solutions of a system of max-type

difference equations of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87–100

Nuha H. Hamada, Adel G. Naoum∆∗(H) operators and the hyperinvariant subspace problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 101–111Mohammad F. Al-Jamal, E. A. RawashdehReconstructing the diffusion coefficient in fractional diffusion equations . . . . . . . . . . . . . . 112–123

M.N.I. Khan, G.A. Ansari, Z.A. AdhoniStructures in a differentiable manifold and their applications

to the tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124–133

Amira A. Ishan, Meraj Ali KhanWarped product submanifolds of a generalized Sasakian space form

admitting nearly cosymplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134–149

A.V.S.N. MurtyStructures on Galois connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150–154

T. Nasri, B. Mashayekhy, H. TorabiOn exact sequences of the rigid fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155–162

N.B. Okelo, P.O. MogotuOn norm inequalities and orthogonality of commutators of derivations . . . . . . . . . . . . . . . 163–169

E. Poudineh, M. Rostamy-Malkhalifeh, A. Payan, A. NouraA fully fuzzy DEA approach for network cost efficiency measurement

based on ranking functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170–187

Mamta Rani, Pammy ManchandaNumerical solution of nonlinear oscillatory differential equations

using shifted second kind Chebyshev wavelet method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188–197

A.A. Azzam, S.S. Hussein, H. Saber OsmanCompactness of topological spaces with grills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198–207

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020 v

A.N. Sebandal, J.P. VilelaFree commutative B-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208–223

M.A. Shakhatreh, T.A. QawasmehAssociativity of max-min composition of three fuzzy relations . . . . . . . . . . . . . . . . . . . . . . . . 224–228

Faiza Shujat, Shahoor Khan, Abu Zaid AnsariOn centralizers and multiplicative generalized derivations of semiprime ring . . . . . . . . . 229–237

Rajinder Sharma, Deepti ThakurSome fixed points theorems in intuitionistic Menger spaces. . . . . . . . . . . . . . . . . . . . . . . . . . .238–249

Xiaoxia Zhang, Yanhua YangNowhere-zero 3-flows in 4-connected simple graphs with independence number 3 . . . . . .250–264

Ping Song, Xiaolong Xin, Juntao WangModal operators on equality algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265–281

Nashat Faried, Hany A. El-Sharkawy, Moustafa M. ZakariaMetric projection in countably seminormed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282–290

Issam Alhadid, Khalid Kaabneh, Hassan Tarawneh, Aysh AlhroobInvestigation of simulated annealing components to solve the university

course timetabling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291–301

Jamil Amir Ali Al-HawasySolvability for continuous classical optimal control associated with

triple hyperbolic boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302–318

Mahmoud H. Darassi, Yousef Abu HourResidual power series technique for solving Fokker-Planck equation . . . . . . . . . . . . . . . . . . 319–332

H. Qawaqneh, H. Alsamir, H. Aydi, M.S. Md Noorani, W. ShatanawFixed point results in α− η-complete metric spaces via w-distances . . . . . . . . . . . . . . . . . . 333–347Hardi N. Aziz, Halgwrd M. Darwesh, Adil K. JabarNew independent paracompact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348–356

A.B.M. Basheer, F. Ali, M.L. AlotaibiOn a maximal subgroup of the Conway group Co3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357–372

Barbora Bat́ıková, Tomáš Kepka, Petr NěmecMinimal left ideals in semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373–387

Barbora Bat́ıková, Tomáš Kepka, Petr NěmecMinimal left ideals in some endomorphism semirings of semilattices . . . . . . . . . . . . . . . . . 388–402

Tanmay Biswas, Ritam BiswasOn some (p, q)-φ relative Gol’dberg type and (p, q)-φ relative Gol’dberg weak

type based growth properties of entire functions of several complex variables . . . . . .403–414

Tanmay Biswas, Ritam BiswasA note on relative φ-order and relative φ-lower order of entire functions

of several complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415–430

Chao WeiOn drift parameter estimation for mean-reversion type nonlinear

nonhomogeneous stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431–438

Marzieh Moradi DaleniA hybrid DESA-MOLP method for finding most preferred solution . . . . . . . . . . . . . . . . . . . 439–448

De-Xue LiPrime-valent one-regular graphs of order 8p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449–453

De-Xue LiPrime-valent one-regular graphs of order 16p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454–459

vi ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

S.J. Aravindan, D. Sasikala, A. DivyaVisualization of cordial graph in human excretion track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460–469

Youcef DjenaihiStochastic optimal control model practices in development, finance

and the industrial production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .470–482

Diana Ahmad, Fang LiOrbit-maximal green sequences and general-maximal green sequences . . . . . . . . . . . . . . . . .483–498

B.-Y. XI, D.-D. Gao, F. QiIntegral inequalities of Hermite-Hadamard type for (α, s)-convex

and (α, s,m)-convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499-510

Xiangguang Dai, Yingyin Tao, Wei Zhang, Yuming FengRobust non-negative matrix factorization for subspace learning . . . . . . . . . . . . . . . . . . . . . . .511–520

H. Shamsan, S. Latha, B.A. FrasinConvolution conditions for q-Sakaguchi-Janowski type functions . . . . . . . . . . . . . . . . . . . . . 521–529

Germina K. Augusthy, Gency Joseph, L. Benedict Michael RajCharacterization of 1-uniform dcsl graphs and learning graphs . . . . . . . . . . . . . . . . . . . . . . . 530–537

G.M. GharibSolutions of nonlinear equations to describe physical models in plasma . . . . . . . . . . . . . . . 538–546

H.-P. Yin, J.-Y. Wang, B.-N. GuoIntegral inequalities of Hermite-Hadamard type for extended

(s,m)-GA-ε-convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547–557

Haider Jebur AliOn Nc-continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558–566

Hongliang Zuo, Fazhen JiangMore accurate Young, Heinz-Heron mean and Heinz inequalities

for scalar and matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567–575

J. Tayyebi, E. HosseinzadehPolynomial form fuzzy numbers and their application in linear

programming with fuzzy variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576–588

Nadia Kadum Humadi, Haider Jebur AliCertain types of functions by using supra ω̂-open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589–601

Phakawat Mosrijai, Aiyared IampanHesitant fuzzy soft sets over UP-algebras by means of anti-type . . . . . . . . . . . . . . . . . . . . . . 602–620

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viii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

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ixITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

Exchanges

Up to December 2015 this journal is exchanged with the following periodicals:

1. Acta Cybernetica - Szeged H 2. Acta Mathematica et Informatica Universitatis Ostraviensis CZ 3. Acta Mathematica Vietnamica – Hanoi VN 4. Acta Mathematica Sinica, New Series – Beijing RC 5. Acta Scientiarum Mathematicarum – Szeged H 6. Acta Universitatis Lodziensis – Lodz PL 7. Acta Universitatis Palackianae Olomucensis, Mathematica – Olomouc CZ 8. Actas del tercer Congreso Dr. Antonio A.R. Monteiro - Universidad Nacional del Sur Bahía Blanca AR 9. AKCE International Journal of Graphs and Combinatorics - Kalasalingam IND10. Algebra Colloquium - Chinese Academy of Sciences, Beijing PRC 11. Alxebra - Santiago de Compostela E 12. Analele Ştiinţifice ale Universităţii “Al. I Cuza” - Iaşi RO

13. Analele Universităţii din Timişoara - Universitatea din Timişoara RO

14. Annales Academiae Scientiarum Fennicae Mathematica - Helsinki SW 15. Annales de la Fondation Louis de Broglie - Paris F 16. Annales Mathematicae Silesianae – Katowice PL 17. Annales Scientif. Université Blaise Pascal - Clermont II F 18. Annales sect. A/Mathematica – Lublin PL 19. Annali dell’Università di Ferrara, Sez. Matematica I 20. Annals of Mathematics - Princeton - New Jersey USA 21. Applied Mathematics and Computer Science -Technical University of Zielona Góra PL 22. Archivium Mathematicum - Brnö CZ 23. Atti del Seminario di Matematica e Fisica dell’Università di Modena I 24. Atti dell’Accademia delle Scienze di Ferrara I 25. Automatika i Telemekhanika - Moscow RU 26. Boletim de la Sociedade Paranaense de Matematica - San Paulo BR 27. Bolétin de la Sociedad Matemática Mexicana - Mexico City MEX 28. Bollettino di Storia delle Scienze Matematiche - Firenze I 29. Buletinul Academiei de Stiinte - Seria Matem. - Kishinev, Moldova CSI 30. Buletinul Ştiinţific al Universităţii din Baia Mare - Baia Mare RO

31. Buletinul Ştiinţific şi Tecnic-Univ. Math. et Phyis. Series Techn. Univ. - Timişoara RO

32. Buletinul Universităţii din Braşov, Seria C - Braşov RO

33. Bulletin de la Classe de Sciences - Acad. Royale de Belgique B 34. Bulletin de la Societé des Mathematiciens et des Informaticiens de Macedoine MK 35. Bulletin de la Société des Sciences et des Lettres de Lodz - Lodz PL 36. Bulletin de la Societé Royale des Sciences - Liege B 37. Bulletin for Applied Mathematics - Technical University Budapest H 38. Bulletin Mathematics and Physics - Assiut ET 39. Bulletin Mathématique - Skopje Macedonia MK 40. Bulletin Mathématique de la S.S.M.R. - Bucharest RO 41. Bulletin of the Australian Mathematical Society - St. Lucia - Queensland AUS 42. Bulletin of the Faculty of Science - Assiut University ET 43. Bulletin of the Faculty of Science - Mito, Ibaraki J 44. Bulletin of the Greek Mathematical Society - Athens GR 45. Bulletin of the Iranian Mathematical Society - Tehran IR 46. Bulletin of the Korean Mathematical Society - Seoul ROK 47. Bulletin of the Malaysian Mathematical Sciences Society - Pulau Pinang MAL 48. Bulletin of Society of Mathematicians Banja Luka - Banja Luka BiH 49. Bulletin of the Transilvania University of Braşov - Braşov RO

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x ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

55. Classification of Commutative FPF Ring - Universidad de Murcia E 56. Collectanea Mathematica - Barcelona E 57. Collegium Logicum - Institut für Computersprachen Technische Universität Wien A 58. Colloquium - Cape Town SA 59. Colloquium Mathematicum - Instytut Matematyczny - Warszawa PL 60. Commentationes Mathematicae Universitatis Carolinae - Praha CZ 61. Computer Science Journal of Moldova CSI 62. Contributi - Università di Pescara I 63. Cuadernos - Universidad Nacional de Rosario AR 64. Czechoslovak Mathematical Journal - Praha CZ 65. Demonstratio Mathematica - Warsawa PL 66. Discussiones Mathematicae - Zielona Gora PL 67. Divulgaciones Matemáticas - Universidad del Zulia YV 68. Doctoral Thesis - Department of Mathematics Umea University SW 69. Extracta Mathematicae - Badajoz E 70. Fasciculi Mathematici - Poznan PL 71. Filomat - University of Nis SRB 72. Forum Mathematicum - Mathematisches Institut der Universität Erlangen D 73. Functiones et Approximatio Commentarii Mathematici - Adam Mickiewicz University L74. Funkcialaj Ekvaciaj - Kobe University J 75. Fuzzy Systems & A.I. Reports and Letters - Iaşi University RO

76. General Mathematics - Sibiu RO 77. Geometria - Fasciculi Mathematici - Poznan PL 78. Glasnik Matematicki - Zagreb CRO 79. Grazer Mathematische Berichte – Graz A 80. Hiroshima Mathematical Journal - Hiroshima J 81. Hokkaido Mathematical Journal - Sapporo J 82. Houston Journal of Mathematics - Houston - Texas USA 83. IJMSI - Iranian Journal of Mathematical Sciences & Informatics, Tarbiat Modares University, Tehran IR 84. Illinois Journal of Mathematics - University of Illinois Library - Urbana USA 85. Informatica - The Slovene Society Informatika - Ljubljana SLO 86. Internal Reports - University of Natal - Durban SA 87. International Journal of Computational and Applied Mathematics – University of Qiongzhou, Hainan PRC 88. International Journal of Science of Kashan University - University of Kashan IR 89. Iranian Journal of Science and Technology - Shiraz University IR 90. Irish Mathematical Society Bulletin - Department of Mathematics - Dublin IRL 91. IRMAR - Inst. of Math. de Rennes - Rennes F 92. Israel Mathematical Conference Proceedings - Bar-Ilan University - Ramat -Gan IL 93. Izvestiya: Mathematics - Russian Academy of Sciences and London Mathematical Society RU 94. Journal of Applied Mathematics and Computing – Dankook University, Cheonan – Chungnam ROK 95. Journal of Basic Science - University of Mazandaran – Babolsar IR 96. Journal of Beijing Normal University (Natural Science) - Beijing PRC 97. Journal of Dynamical Systems and Geometric Theory - New Delhi IND 98. Journal Egyptian Mathematical Society – Cairo ET 99. Journal of Mathematical Analysis and Applications - San Diego California USA 100. Journal of Mathematics of Kyoto University - Kyoto J 101. Journal of Science - Ferdowsi University of Mashhad IR 102. Journal of the Bihar Mathematical Society - Bhangalpur IND 103. Journal of the Faculty of Science – Tokyo J 104. Journal of the Korean Mathematical Society - Seoul ROK 105. Journal of the Ramanujan Mathematical Society - Mysore University IND 106. Journal of the RMS - Madras IND 107. Kumamoto Journal of Mathematics - Kumamoto J 108. Kyungpook Mathematical Journal - Taegu ROK 109. L’Enseignement Mathématique - Genève CH 110. La Gazette des Sciences Mathématiques du Québec - Université de Montréal CAN 111. Le Matematiche - Università di Catania I 112. Lecturas Matematicas, Soc. Colombiana de Matematica - Bogotà C 113. Lectures and Proceedings International Centre for Theorical Phisics - Trieste I 114. Lucrările Seminarului Matematic – Iaşi RO

115. m-M Calculus - Matematicki Institut Beograd SRB 116. Matematicna Knjiznica - Ljubljana SLO

xi ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44-2020

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141. Monthly Bulletin of the Mathematical Sciences Library – Abuja WAN 142. Nagoya Mathematical Journal - Nagoya University,Tokyo J 143. Neujahrsblatt der Naturforschenden Gesellschaft - Zürich CH 144. New Zealand Journal of Mathematics - University of Auckland NZ 145. Niew Archief voor Wiskunde - Stichting Mathematicae Centrum – Amsterdam NL 146. Nihonkai Mathematical Journal - Niigata J 147. Notas de Algebra y Analisis - Bahia Blanca AR 148. Notas de Logica Matematica - Bahia Blanca AR 149. Notas de Matematica Discreta - Bahia Blanca AR 150. Notas de Matematica - Universidad de los Andes, Merida YV 151. Notas de Matematicas - Murcia E 152. Note di Matematica - Lecce I 153. Novi Sad Journal of Mathematics - University of Novi Sad SRB 154. Obzonik za Matematiko in Fiziko - Ljubljana SLO 155. Octogon Mathematical Magazine - Braşov RO

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179. Publications of Serbian Scientific Society - Beograd SRB 180. Publikacije Elektrotehnickog Fakulteta - Beograd SRB 181. Pure Mathematics and Applications - Budapest H 182. Quaderni di matematica - Dip. to di Matematica – Caserta I 183. Qualitative Theory of Dynamical Systems - Universitat de Lleida E 184. Quasigroups and Related Systems - Academy of Science - Kishinev Moldova CSI 185. Ratio Mathematica - Università di Pescara I 186. Recherche de Mathematique - Institut de Mathématique Pure et Appliquée Louvain-la-Neuve B 187. Rendiconti del Seminario Matematico dell’Università e del Politecnico – Torino I 188. Rendiconti del Seminario Matematico - Università di Padova I 189. Rendiconti dell’Istituto Matematico - Università di Trieste I 190. Rendiconti di Matematica e delle sue Applicazioni - Roma I 191. Rendiconti lincei - Matematica e applicazioni - Accademia Nazionale dei Lincei I 192. Rendiconti Sem. - Università di Cagliari I 193. Report series - Auckland NZ 194. Reports Math. University of Stockholm - Stockholm SW 195. Reports - University Amsterdam NL 196. Reports of Science Academy of Tajikistan – Dushanbe TAJ 197. Research Reports - Cape Town SA 198. Research Reports - University of Umea - Umea SW 199. Research Report Collection (RGMIA) Melbourne AUS 200. Resenhas do Instituto de Matemática e Estatística da universidadae de São Paulo BR 201. Review of Research, Faculty of Science, Mathematics Series - Institute of Mathematics University of Novi Sad SRB 202. Review of Research Math. Series - Novi Sad YN 203. Revista Ciencias Matem. - Universidad de la Habana C 204. Revista Colombiana de Matematicas - Bogotà C 205. Revista de Matematicas Aplicadas - Santiago CH 206. Revue Roumaine de Mathematiques Pures et Appliquées - Bucureşti RO

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228. Studii şi Cercetări Ştiinţifice, ser. Matematică - Universitatea din Bacău RO

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ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 44–2020 (1–6) 1

Two-level secret sharing schemes based on reverse super edgemagic labelings

Md. ShakeelJNTUHHyderabad, TelanganaIndia

Sharief BashaVIT UnversityVellore, Tamil NaduIndia

Raja Das∗

VIT Unversity

Vellore, Tamil Nadu

India

[email protected]

Abstract. In this paper, we propose two-level secret sharing scheme based on areverse edge magic labeling of star graphs. It is a scheme which creates two types ofhierarchical sets. The first set contains share that are more powerful than the share inthe second set. We build banking secret sharing scheme that will share a secret amongone bank manager and several sets of authorized staff members.

Keywords: two-level secret sharing schemes, edge-magic total labelling.

1. Introduction

In 1979, Blakley [3], Shamir [7], and Chaum [5] introduced the notion of secretsharing scheme. A secret sharing scheme is a method of allocating a secret Samong a finite set of participants P = {p1, p2, · · · , pn} in such a way that ifthe participants in A ⊆ P are capable to know the secret, then their partialinformation by pooling together, they can reconstruct the secret S; but anyB ⊆ P , which is not qualified to know S, cannot reconstruct the secret. Thekey S which is chosen by a special participant d, called a dealer, and it usuallyassumed that d /∈ P . The share means that dealer gives partial information toeach participant which is a tool to reveal the secret S. An access structure Γis the family of all the subsets of participants that are able to reconstruct thesecret. The sets belonging to the access structure Γ are called authorized setsand those not belonging to the access structure are termed as unauthorized sets.

A two-level secret sharing scheme is a scheme which produces two kindsof hierarchical sets. The first set (the highest rank) contains shares that are

∗. Corresponding author

2 MD. SHAKEEL, SHARIEF BASHA and RAJA DAS

more powerful (important) than the shares in the second set. In our previouspaper, we construct a secret sharing scheme based on reverse edge magic graphlabeling. In this paper, we continue our work to plan a two-level secret sharingscheme based on reverse edge magic graph labeling. We provide two differenttypes of such schemes, with two different possible applications.

The first scheme distributes shares of a secret among two sets. The first set(the highest rank set) contains a single person s0, called a supervisor, and thesecond set contains a number of chosen people. In the first scheme, the accessstructure Γ is the family of all sets of the form {s0, p} where p belongs to thesecond set. We call the first scheme the supervision al secret sharing scheme.

In the proposed second scheme, the first set (the highest rank set) containsa single person s0. The second set S2 contains k departments, namely S2 ={D1, D2, · · · , Dk} where each department Di consists of di authorised people.In the second scheme ,the access structure Γ in this second scheme is the familyof all sets of the form {s0, x1, x2, · · · , xk} where xi ∈ Di. We name this schemethe departmental secret sharing scheme. These two secret sharing schemes areconstructed by reverse edge-magic labeling. The definitions of reverse edge-magic labeling and its related results will be existing in Section II. The proposedschemes will be enlightened in Section III. A conclusion is located in Section IV.

2. Basic theory

In this paper, we considered finite and simple graphs and used general referencefor graph-theoretic ideas in [?] . The graph G with the vertex-set V (G) andthe edge-set E(G) is said to be a reverse edge-magic (REM) labelingthen theone-to-one mapping

f : V (G) ∪ E(G)→ {1, 2, 3, · · · , |V (G) + E(G)|}

satisfying the property that there exists an integer k such that

f(xy)− {f(x) + f(y)} = k

for each edge xy in G. We call f(xy) − {f(x) + f(y)} = k the reverse edgedifference of edge xy, and k the reverse magic constant of graph G. In partic-ular, if f(V (G)) = {1, 2, · · · , |V (G)|} then f is called reverse super edge-magiclabeling. A graph is called reverse(super) edge-magic if it admits any reverse(super) edge-magic labeling.

Venkata Ramana et al [9] introduced and studied the notion of reverse edge-magic graphs with a different name, i.e., graphs with reverse magic valuations,while the term of reverse super edge-magic graphs was firstly introduced bythem. They showed that a star Sn+1 = K1, n is the only complete bipartitegraph which is reverse super edge-magic total. They also showed that any oddcycle is reverse super edge-magic, but every wheel is not. Since then, some ofauthors have studied reverse(super) edge-magic properties in graphs, see, for

TWO-LEVEL SECRET SHARING SCHEMES BASED ON REVERSE SUPER EDGE MAGIC ... 3

Figure 1: Reverse Super Edge-Magic Labeling

instances, [8, 9]. Figure 1 shows a reverse super edge-magic labeling on graphtree on 6 vertices with the reverse magic constant k = 2.

In the following section, we suggest two schemes for secret sharing basedon an reverse super edge-magic labeling on a star Sn. The distribution andreconstruction algorithms in this schemes is calculated to work based on ourknowledge of reverse super edge-magic labeling, in specific on stars .

3. Proposed schemes

Since the notion of secret sharing scheme introduced, various different typesof secret sharing schemes have been proposed (by many authors). They usedmathematical structures, such as vector spaces, polynomial, block design, roomsquares [5], and latin squares [7], to design secret sharing schemes. In this paper,we proposed two schemes for secret sharing based on edge-magic total labelingon graphs.

3.1 Supervisional secret sharing scheme

In supermarkets or banks, when a cashier or teller wants to change or cancelan (important) transaction, on their computer, they will need their supervisorapproval to do such a thing. The supervisor will enter a password and thenthe cashier or teller will complete the entry by his/her own password. In thissituation, we can see that the supervisor has more power password then thecashier or teller.

Based on this situation, we build supervisional secret sharing scheme thatwill share a secret among one supervisor and set of participants under his/hersupervision (staffs). In the following we propose an algorithm for this purpose.

Algorithm 3.1

Distribution Algorithm

Input:

• The length of bank manager’s share, n1 ( n1 even).


• The length of bank employee’s share, n2 ( n2 even).

• The number of bank employee, r.

Steps:

1) Build a reverse edge magic labelling on star Sn.

2) Select the label randomly for the center from the integer set (1, n+1, 2n+1), then we have a particular sum k.

3) Compute the labelling.

4) Set k as a secret

Output:

• The share for the bank manager, s0 ( s0 even).

• The share for each his/her bank employee, pi for i = 1, 2, · · · , r.

Reconstruction AlgorithmInput:

1) The share of the supervisor, s0 ( s0 even)

2) The share of one of the staff, L.

3) The size of the star, n (kept by the system)

Steps

1) Built the reverse edge magic number as H = f(xy)− {f(x) + f(y)}.

2) If H is the reverse edge magic constant k, then the secret is revealed,otherwise secret is not revealed.

Output: The secret is revealed or not.In this scheme, the secret is an edge-magic total labeling on graph Sn. The

secret will be shared to a bank manager and his/her employee.

3.2 Departmental secret sharing scheme

In institutions with several departments, in some situation, to do an agreement,it will need some approval, those are from the head of the institution and facultyfrom each department. They give their approval by signing the agreement.

Based on this, we build departmental secret sharing scheme that will share asecret among one head of institution and several faculties. These sets representthe departments and the faculties be the authorized representatives.

Algorithm 3.2Distribution Algorithm (with two departments)Input:

TWO-LEVEL SECRET SHARING SCHEMES BASED ON REVERSE SUPER EDGE MAGIC ... 5

• The length of share for the director.

• The length of share for each HOD in Department A.

• The length of share for each HOD in Department B.

• The number of Faculties in Department A.

• The number of Faculties in Department B.

Steps:

1) Build a reverse edge magic labelling on Tree ⟨K1,n1 ,K1,n2⟩

2) Select , at random, a reverse edge magic labelling k on the Tree ⟨K1,n1 ,K1,n2⟩

3) Set λ as a secret.

Output:

• descriptionThe share for the director.

• The share for Faculty in Department A.

• The share for Faculty in Department B.

Reconstruction AlgorithmInput:

The share of Faculty in Department A.

The share of Faculty in Department B.

The share of Director.

The size of a tree

Steps

1) Define k = f(xy)− {f(x) + f(y)}

2) If k = λ then secret is revealed, then otherwise secret is not revealed.

4. Conclusion

This paper proposed two level schemes of secret sharing based on reverse edge-magic labeling on graph G. In these methods, the secret is a chosen reverse edge-magic labeling, in particular, on star Sn and tree ⟨K1,n1 ,K1,n2⟩. To distributethe shares, the algorithms work based on a reverse edge magic labeling. Thereconstruction algorithms for these schemes are based on our knowledge on kfor star/ Tree.


References

[1] E.T. Baskoro, M. Miler Slamin, W.D. Wallis Edge magic total labelings ,Australasian Journal of Combinatorics, 22 (2000), 177-190.

[2] E.T. Baskoro, R. Simanjuntak, M.T. Adithia, Secret Sharing scheme basedon magic labeling, Proc. of the 12th National Conference on Mathematics,2004, 23-27.

[3] G.R. Blakley, Safeguarding cryptographic keys, Proc. AFIPS, New York, 48(1979), 313-317.

[4] G.R. Chaudhry, H. Ghodosi, J. Seberry, Perfect secret sharing schemesbased on room squares.

[5] D. Chaum, Computer systems established, maintained, and trusted by mutu-ally suspicious groups, Memorandum No. UCB/ERL M179/10, Universityof California Berkeley, CA, 1979.

[6] E.D. Karnin, J.W. Greene, M.E. Hellman, On Secret Sharing Systems,IEEE Trans. Inf. Th., vol. IT-29(1), 1983, 35-41.

[7] A. Shamir, How to Share a Secret, Comm. ACM, 22 (1979), 612-613.

[8] Md. Shakeel, S. Sharief Basha, K.J. Sarma Smieee K.J., Algorithms for con-structing Reverse edge magic labelling of complete bipartite graphs, GlobalJournal of Pure and Applied Mathematics, 12, 707-710.

[9] S. Venkata Ramana, S. Sharief Basha, Reverse Super edge magic labellingof a graph, PhD Thesis, 2009.

Accepted: 13.12.2017


Edge maximal W7-free graphs

Ahmad AL-RhayyelDepartment of Mathematics

Yarmouk University

Irbid

Jordan

[email protected]

Abstract. Extremal graph theory is one of the most important subjects in graphtheory. In this paper we give an upper bound to the number of edges of graphs whichare W7-free. In fact we prove that if G is a graph on n-vertices which is W7-free, then

|E(G)| ≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2.

Keywords: extremal graphs, wheels.

1. Introduction

In this paper we only consider finite, undirected simple graphs. The vertex setof a graph G is denoted by V (G) and the edge set is denoted by E(G). A cycleon n-vertices is denoted by Cn. The degree of a vertex v in V (G) is denotedby d(v). Moreover, we denote the minimum degree of vertices of G by δ(G)and the maximum degree by △(G). Also, we let ε(G) and υ(G) denote |E(G)|and |V (G)|, resp. The graph formed by taking a cycle Cn−1 on n − 1 vertices,and a vertex u /∈ V (Cn−1) by joining u to each vertex of Cn−1 is called a wheeland is denoted by Wn. We say that a graph G is H-free if G does not containa subgraph isomorphic to the graph H as a subgraph. If u ∈ V (G), then wedenote by NG(u) or simply by N(u) the set of vertices of G adjacent to u andwe denote by N [u] the set N(u)

∪u. Two nonadjacent vertices u, v in V (G) are

called twins if N(u) = N(v). If H1 and H2 are vertex disjoint subgraphs of agraph G, then we let E(H1 ∪H2) = {xy ∈ E(G) : x ∈ V (H1), y ∈ V (H2)} andε(H1,H2) = |E(H1 ∪H2)|. Let F be a set of graphs and n be a positive integerand let G(n,F) denote the class of non-bipartite F-free graphs on n-vertices,and

f(n,F) = max{ε(G) : G ∈ G(n,F)}.

The branch of graph theory which aims at finding the values of the functionf(n,F) is extremal graph theory which was initiated by Turan’s [16] in 1941.Many researchers have studied these problems; for example, see [1]–[5],[9],[11],[13]–[17]. Next, we state some of the results found.

8 AHMAD AL-RHAYYEL

Theorem 1.1 (Erdös-stone-simonovite). Let F be any finite set of graphs andlet r be the minimum number of chromatic number of F in F , then

f(n,F) = (1− 1r − 1

)

(n

2

)+ o(n2).

Theorem 1.2 ([15]). Let G be a graph on n-vertices such that F is wheel free,

then ε(G) ≤ ⌊n24 ⌋+ ⌊n+14 ⌋.

Al-Rhayel et al. in [1] proved that f(n,W5) ≤ ⌊n−24 ⌋+ ⌊s4⌋ for n ≥ 3 where

s = n if n ̸= 4k + 2 and s = n− 1 if n = 4k + 2 and

f(n,W6) ≤ ⌊n2

3⌋

for n ≥ 6.

Theorem 1.3 ([7]). Let G be a graph on n-vertices with ε(G) > ⌊n24 ⌋, then Gcontains a cycle of every length L with 3 ≤ L ≤ ⌊n+32 ⌋.

Haggkvist et al. [12] proved that: f(n,Cr) ≤ ⌊ (n−2)2)

4 ⌋+ 1, for all r. Jia [14]proved that f(n,C5) ≤ ⌊ (n−2)

2

4 ⌋+ 3 for n ≥ 9. Bataineh et al. [5] proved that

f(n, θ7) ≤ ⌊(n− 1)2

4⌋+ 3.

Jaradat et al. [13] proved that f(n, θ2k+1) ≤ ⌊ (n−2)2

4 ⌋+ 3.Finally, we state Turan’s theorem [16].

Theorem 1.4. Let G be a graph on n-vertices without a k-clique, then

|E(G)| ≤ (k − 2)n2

2(k − 1).

2. Edge maximal W7-free graphs on n(≤ 10)-vertices

Throughout this section, let G be a graph on n-vertices (n ≤ 10); clearly, ifn ≤ 6, then G has no W7 as a subgraph, so we will only deal with the casesn = 7, 8, 9, and 10. Most of this section’s results can be found in [2].

Theorem 2.1. Let G be a graph on 7-vertices if G is W7-free then |E(G)| ≤ 17.

Proof. Let {x1, · · · , x7} be the vertices of G: if G contains K6 as a subgraph,let {x1, · · · , x6} be the vertices of K6; hence, x7 is the seventh vertex of G, thenclearly x7 can be adjacent to at most two vertices of K6, and hence |E(G)| ≤15 + 2 = 17. Figure 1 below shows such a graph: notice that the addition ofany new edge to this graph produces W7 as a subgraph of G.

If G does not contain K6 as a subgraph, but contains K5: let {x1, · · · , x5}be the vertices of K5. Now x6 ∈ V (G), if |NK5(x6)| = 5, then G contains K6

EDGE MAXIMAL W7-FREE GRAPHS 9

which is a contradiction, hence |NK5(x6)| ≤ 4. In fact, it is easy to check that wemay assume NK5(x6) = {x1, · · · , x4}, x7 ∈ V (G). Figure 2 shows such a graphG, in which the addition of any new edge produces K6 or W7 as a subgraphof G, and clearly |E(G)| = 16 ≤ 17. If G does not contain K5 but containsK4 as a subgraph of G, let {x1, · · · , x4} be the vertices of K4, x5 ∈ V (G), ifNK4(x5) = {x1, · · · , x4}, then G contains K5, which is a contradiction. Hence,we may assume that NK4(x5) = {x1, x2, x3} and let G1 be the resulting graphon {x1, · · · , x5}. If |NG1(x6)| = 5, then G[x1, · · · , x6] contains K5 as a sub-graph, also a contradiction. Hence |NG1(x6)| ≤ 4, and clearly, we may assume|NG1(x6)| = {x1, x2, x4, x5}. Let G2 be the resulting graph on {x1, · · · , x6}.Now x7 ∈ V (G), and we may assume NG2(x7) = {x3, x4, x5, x6}. Figure 3 dis-plays such a graph G, where the addition of any new edge to G produces a K5or a W7 as subgraph of G, hence |E(G)| = 6 + 3 + 8 = 17.

Theorem 2.2. Let G be a graph on 8 vertices, if G is W7–free, then |E(G)| ≤21.

Proof. Let {x1, · · · , x8} be the vertices of G. If G contains K6 as a subgraphof G, let {x1, · · · , x6} be the vertices of K6. x7, x8 ∈ V (G) and, as above (casen = 7), |NK6(xi)| ≤ 2, i ∈ {7, 8}. Clearly, we may assume NK6(x7) = {x1, x2},NK6(x8) = {x3, x4}, and x7x8 ∈ E(G). Figure 4 shows such a graph G, wherethe addition of any new edge to G produces W7 as a subgraph of G, hence|E(G)| = 20. If G does not contain K6 but contains K5 as a subgraph, let{x1, · · · , x5} be the vertices K5. x6 ∈ V (G), if |Nk5(x6)| = 5, then we get K6as a subgraph of G, hence |NK5(x6)| ≤ 4, and as in case n = 7, we may assumeNK5(x6) = {x1, x2, x3, x4} and NG1(x7) = {x1, x6}, where G1 is the resultinggraph on {x1, · · · , x6}. Let H be the resulting graph on x1, · · · , x7. Now x8 ∈V (G) and clearly we may assume NH(x8) = {x1, x6}, and x7x8 ∈ E(G). Noticethat Figure 5 displays such a graph G, where the addition of any new edge tothis graph produces K6 or W7 as a subgraph of G; hence |E(G)| = 19. If Gdoes not contain K5 but contains K4 as a subgraph, let {x1, · · · , x4} be thevertices of K4. Now x5, x6, x7 ∈ V (G) and by analysis similar to that used forcase n = 7, we may assume NK4(x5) = {x1, x2, x3}, NG1(x6) = {x1, x2, x4, x5}and NG2(x7) = {x3, x4, x5, x6}, where G1 is the resulting graph on {x1, · · · , x5},and G2 is the resulting graph on {x1, · · ·x6}. Let G3 be the resulting graph on{x1, · · · , x7} (see figure 6 below). x8 ∈ V (G), and one can check that we mayassume NG3(x8) = {x3, x4, x5, x6}. Figure 7 shows such a graph G, where theaddition of any new edge to this graph produces K5 or W7 as a subgraph of G,hence |E(G)| = 21.

Theorem 2.3. Let G be a graph on 9-vertices if G is W7-free then

|E(G)| ≤ 25.

Proof. Let {x1, · · · , x9} be the vertices of G. If G contains K6 as a subgraph,let {x1, · · · , x6} be the vertices of G. Notice that if v ∈ {x7, x8, x9}, then

10 AHMAD AL-RHAYYEL

|E(v,K6)| ≤ 2, and if B = G[x7, x8, x9], then |E(B)| ≤ 3, hence |E(G)| ≤15 + 6 + 3 = 24. If G does not contain K6 but contains K5 as a subgraph,let {x1, · · · , x5} be the vertices of K5. x6 ∈ V (G). Clearly, we may assumeNK5(x6) = {x1, x2, x3, x4}. Let G1 be the resulting graph on {x1, · · · , x6}. Now{x7, x8, x9} ⊆ V (G); clearly, |NG1(xi)| ≤ 2, for i = 7, 8, 9 and we may assumex7x8, x7x9 and x8x9 are edges in G; hence, |E(G)| ≤ 10 + 4 + 6 + 3 = 23. Figure8 below shows such a graph G, where the addition of any new edge producesK6 or W7 as a subgraph of G.

If G does not contain k5 but contains K4 as a subgraph, then by analysisas in previous cases, one can easily check that the graph G given in Figure 9ensures that G contains K4 and the addition of any new edge to G produces K5or W7 as a subgraph of G, and |E(G)| ≤ 6 + 3 + 16 = 25.

Theorem 2.4. Let G be a graph on 10-vertices, if G is W7–free then

|E(G)| ≤ 30.

Proof. Let {x1, · · · , x10} be vertices of G. If G contains K6 as a subgraph, thenlet {x1, · · · , x6} be the vertices of K6. Notice that if v ∈ {x7, x8, x9, x10}, then|E(v,K6)| ≤ 2 and if B = G[x7, x8, x9, x10], then |E(B)| ≤ 6, hence |E(G)| ≤15 + 8 + 6 = 29. If G does not contain K6 but contains K5 as a subgraph,let {x1, · · · , x5} be the vertices of K5. x6 ∈ V (G), and clearly we may assumeNK5(x6) = {x1, x2, x3, x4}. Let G1 be the resulting graph on {x1, · · · , x6}, andfollowing an analysis like for n = 9, we notice that {x7, x8, x9, x10} ⊆ V (G),and G[x7, x8, x9, x10] = K4 and |NG1(xi)| ≤ 2, i = 7, 8, 9, 10, hence |E(G)| ≤10 + 4 + 8 + 6 = 28. If G does not contain K5 but contains K4 as a subgraph,let {x1, · · · , x4} be the vertices of K4, then following an analysis as in previouscases (for more details consult [2]), we get |E(G)| ≤ 30.

Remark 2.5. The bound in Theorem 2.4 is sharp as illustrated in example 2.1below.

Example 2.1. Let G1 be the graph shown in Figure 10, where each vertex inH1 is joined to each vertex of H2, then clearly G1 has exactly 30 edges and isW7–free. and the addition of any new edge to G1 produces W7 as a subgraphof G1.

3. Main results

Let G be a graph of order n ≥ 11, following Moon [15]. Let Hn be the class ofgraphs obtained by partitioning the vertices of G into two sets, P and Q, suchthat P has ⌊n+12 ⌋ vertices and Q has ⌊

n2 ⌋ vertices, and there are as many edges

joining pairs of vertices in P and Q as consistent with the requirement that notwo of these edges have a vertex in common, and each vertex in P is joined toeach vertex in Q.


Remark 3.1. 1. If two edges, say in P , have a common vertex, say x1x2, x2x3,then let y1y2, y3y4 be two edges in Q, and notice that x2 is adjacent to everyvertex of the 6–cycle (x1y1y2x3y4y3x1). This produces W7 as subgraph of G.

2. Figure 11 below shows a graph G in Hn: clearly G is W7–free andε(G) ≥ ⌊n24 ⌋+ ⌊

n2 ⌋, for all n ≥ 7. So we conclude that:

f(n,W7) ≥ ⌊n2

4⌋+ ⌊n

2⌋.

Lemma 3.2. Let G be a W7–free graph on n–vertices (n ≥ 7) such that K6 isa subgraph of G, then |E(v,K6)| ≤ 2 for all v ∈ V (G−K6).

Proof. Let {x1, x2, ...x6} be the vertices of K6, if |NK6(v)| = 3. Then we mayassume NK6(v) = {x1, x2, x3}; clearly x2 is adjacent to every vertex of the 6–cycle (x1vx3x4x5x6), and hence W7 is produced as a subgraph of G, which is acontradiction.

In Remarks 3.3 and 3.4 below we construct a class of graphs Fn such that ifG belongs to Fn, then G is W7−free and |E(G)| ≥ ⌊n

2

4 ⌋+ ⌊n2 ⌋+ 1.

Remark 3.3. Let G be a graph of order n ≥ 11. Let Fn be the class of graphsobtained by partitioning the vertices of G into two sets P,Q such that P has⌊n+12 ⌋ vertices and Q has ⌊

n2 ⌋ vertices, and there are many edges joining vertices

in P and Q, as consistent with the following requirements.

1. P has no vertices of degree 3.

2. If H has component of P , then H is one of the following: a single vertex,aP2, aC3 or aC4.

3. Q has exactly one edge joining any two vertices of Q and the rest of thevertices in Q are vertices of degree 0.

4. Each vertex in P is joined to each vertex in Q; figure 12 below shows agraph G in the class Fn.

Remark 3.4. 1. Remark 3.3 above shows that a graph G in Fn is W7–free.2. If C5 is a component in P , {x1, x2, x3, x4, x5} be this cycle, and let y1, y2

be the edge in G, then y1 is adjacent to each vertex of the cycle (y2x1x2x3x4x5);hence, W7 is produced as a subgraph of G.

3. If P has a vertex x1 of degree 3, then let N(x1) = {x2, x3, x4}, let y1y2be the edge in G, let y3 be a vertex of degree 0 in G and notice that under theabove construction, x1 is adjacent to every vertex of the cycle (x2y3x4y2x3y1),and this produces W7 as a subgraph of G.

4. The maximum number of edges in P is obtained if all components of Pare either C3’s, C4’s or a combination of both, and hence this implies that eachvertex of P has degree 2, thus

|E(G)| ≥ ⌊n+ 22⌋⌊n

2⌋+ [n+ 1

4] + 1 ≥ ⌊n

2 + 2n+ 1

4⌋+ 1⌊n

2

4⌋+ ⌊n

2⌋+ 1.

12 AHMAD AL-RHAYYEL

Theorem 3.5. Let G be a W7−free graph on n− vertices (n ≥ 10). If Gcontains exactly one copy of K6 as a subgraph, then |E(G)| ≤ ⌊n

2

4 ⌋+ ⌊n+12 ⌋+ 2.

Proof. We use induction on n, for n = 7 see Theorem 2.1. So assume the resultis true for all graphs having less than n−vertices, and assume G is a graph onn−vertices. Let B = G−K6, then B is a graph with |V (B)| < n, hence

|E(B)| ≤ ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 2,

notice that if x ∈ V (B), than E(x,K6) ≤ 2, since otherwise W7 is produced asa subgraph of G; Thus |E(B,K6) ≤ 2(n− 6), and

|E(G)| ≤ |E(K6)|+ |E(B,K6)|+ |E(B)|,

≤ 15 + 2(n− 6) + ⌊(n− 6)2

4⌋+ ⌊n− 5

4⌋+ 2,

≤ 12 + 8n+ n2 − 12n+ 36 + n− 5 + 8

4,

=n2 + 2n+ 2− 5n+ 49

4,

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2, if n ≥ 10.

Theorem 3.6. Let G be a W7−free graph on n− vertices (n ≥ 8). If G containsexactly one copy of K5 as a subgraph, but it does not contain K6, then |E(G)| ≤⌊n24 ⌋+ ⌊

n+12 ⌋+ 2.

Proof. we use induction on n. The result being proved for n = 7, 8, 9, 10 (seeTheorems 2.1, 2.2, 2.3 and 2.4). Assume the result holds for all graphs havingless than n−vertices, and let G is a graph on n−vertices. Let G1 be the graphon 6−vertices shown in Figure 13. Let B1 = G − G1, then |V (B1)| < n, andhence |E(B1)| ≤ ⌊ (n−6)

2

4 ⌋+ ⌊n−52 ⌋+ 2.

If x ∈ V (B1), then clearly E(x,K6)| ≤ 2, as otherwise either K6 or W7 isproduced as a subgraph of G, hence |E(B1,K6)| ≤ 2(n− 6); therefore

|E(G)| ≤ |E(G1)|+ |E(B1)|+ |E(B1,K6)|,

≤ 14 + ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 2 + 2(n− 6),

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2, if n ≥ 8.

Theorem 3.7. Let G be a W7−free graph on n− vertices (n ≥ 10). If Gcontains K4 but it does not contain K5 as a subgraph, then |E(G)| ≤ ⌊n

2

4 ⌋ +⌊n+12 ⌋+ 2.


Proof. we use induction on n (n ≥ 10). Let {x1, . . . , xn} be the vertices ofG, and {x1, . . . , x4} be the vertices of K4. x5 ∈ V (G), hence |NK4(x5)| ≤ 3,otherwise we get K5 as a subgraph of G. Clearly we may assume that NK4(x5) ={x1, x2, x3}. Let G1 be resulting graph on {x1, x2, x3, x4, x5}, note that x6 ∈V (G). If |NG1(x6)| = 5, then K5 is a produced as a subgraph of G, hence|NG1(x6)| ≤ 4, and clearly we may assume NG1(x6) = {x1, x2, x4, x5}. Let G2be a resulting graph on {x1, . . . , x6}, now x7 ∈ V (G), if |NG2(x7)| = 5, then K5is produced as a subgraph of G, hence |NG2(x7)| ≤ 4, and we may assume thatNG2(x7) = {x3, x4, x5, x6}. Let G3 be the resulting graph on {x1 . . . , x7}. Noticethat the addition of any new edge to G3 either produces K5 or W7 as a subgraphof G. Now x8 ∈ V (G), and we may assume that NG3(x8) = {x3, x4, x5, x6}. LetG4 be the resulting graph on {x1, . . . , x8}, and notice that the addition of anynew edge to G4 either produces K5 or W7 as a subgraph of G. Also x9 ∈ V (G),and by analysis as above we conclude that |NG4(x9)| ≤ 4, and may assume thatNG4(x9) = {x1, x2, x7, x8}. Let G5 be the resulting graph on {x1, . . . , x9}, andnotice that the addition of any new edge to G5 (see Figure 14) produces eitherK5 or W7 as a subgraph of G, hence |E(G5)| ≤ 25. Let B = G − G5, hence|V (B)| = n− 9 < n, and by induction hypothesis we have:

|E(B)| ≤ ⌊(n− 9)2

4⌋+ ⌊n− 8

2⌋+ 2.

Note that if x ∈ V (B), then |E(x,G5)| ≤ 4, hence |E(B,K5)| ≤ 4(n − 9), andwe conclude that:

|E(G)| ≤ |E(G5)|+ |E(B)|+ |E(B,G5)|,

≤ 25 + ⌊(n− 9)2

4⌋+ ⌊n− 8

2⌋+ 2 + 4(n− 9),

≤ 25 + ⌊(n− 9)2

4⌋+ ⌊n− 8

2⌋+ 2 + 4n+ 36,

≤ ⌊n2 − 18n+ 81

4⌋+ ⌊2n− 16

4⌋+ 4n− 9,

≤ ⌊n2

4⌋+ ⌊−18n+ 81 + 2n− 16 + 16n− 36

4⌋,

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 27− 2n

4,

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2, if n ≥ 10.

Theorem 3.8. Let G be a graph of order n ≥ 11, if G is W7–free, then

|E(G)| ≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2.

14 AHMAD AL-RHAYYEL

Proof. To prove this theorem we use strong mathematical induction on m,where m is the number of vertex disjoint K5 subgraphs contained in G. Ifm = 0, then done by Theorem 3.7, hence we assume that the result holds forall values less than m. We need to prove that the result holds for m, let Gbe a graph on n–vertices having m vertex disjoint K5 subgraphs. Let K5 be asubgraph of G, {x1, x2, x3, x4, x5} be the vertices of K5, and B = G−K5.Case 1. There exist x6 ∈ V (B) such that x6 is adjacent to all vertices of K5,hence K6 is produced as a subgraph of G. let B1 = G−K6; then ε(B1,K6) ≤ 2,for all x ∈ B1, hence ε(B1,K6) ≤ 2(n− 6), and by induction hypothesis

ε(B1) ≤ ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 1,

hence

ε(G) ≤ ε(B1) + ε(K6, B1) + ε(K6)

≤ ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 1 + 2(n− 6) + 15

≤ ⌊n2 − 12n+ 36

4⌋+ ⌊2n− 10

4⌋+ 2n+ 4

≤ ⌊n2

4⌋ − 3n+ 9 + ⌊n+ 1

2⌋ − 3 + 2n+ 4

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 1 if − n+ 10 ≤ 1

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 1 if n ≥ 9

≤ n2

4⌋+ ⌊n+ 1

2⌋+ 2 if n ≥ 9.

Case 2. There exist x6 ∈ V (B) such that x6 is adjacent to four vertices ofK5; clearly, we may assume NK5(x6) = {x1, x2, x3, x4}. Let H be the resultinggraph on {x1, · · · , x6}, and let B2 = G−H. We claim that ε(y,H) ≤ 2 for ally in V (B2). To prove this claim, assume |NH(y)| = 3, since NH(x5) = NH(x6)and NH(x1) = NH(x2) = NH(x3) = NH(x4); then NH(y) must be one of thefollowing sets: {x1, x2, x3}, {x1, x2, x5}, {x1, x5, x6}, if NH(y) = {x1, x2, x3}.Then, x2 is adjacent to every vertex of the 6–cycle (yx3x6x4x5x1). If NH(y) ={x1, x2, x5}, then x2 is adjacent to every vertex of the 6–cycle (yx1x6x3x4x5).If NH(y) = {x1, x5, x6}, then x1 is adjacent to every vertex of the 6–cycle(yx6x2x3x4x5). In any case, W7 is produced as a subgraph of G and this provesour claim. Now

ε(G) ≤ ε(B2) + ε(H,B1) + ε(H)

≤ ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 1 + 2(n− 6) + 14

≤ ⌊n2

4⌋ − 3n+ 9 + ⌊n+ 1

2⌋ − 3 + 1 + 2n+ 2


≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 1 if − n+ 9 ≤ 1

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 1 if n ≥ 8

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2 if n ≥ 8.

Case 3. There exists a vertex x6 ∈ V (B) such that x6 is adjacent to 3-verticesof K5; clearly, we may assume these vertices to be {x1, x2, x3}. Let H1 be theresulting graph on {x1, · · · , x6}, B3 = G −H1 and y ∈ V (B3). We claim thatε(y,H1) ≤ 3 for all y ∈ V (B3). To prove this claim, notice that if |NH1(y)| = 4,then x6 ∈ NH1(y), since otherwise y is adjacent to 4 vertices of K5, and we goback to case 2 above and since NH1(x1) = NH1(x2) = NH1(x3) andNH1(x4) =NH1(x5) then we may assume that NH1(y) to be one of the following sets:{x1, x2, x3, x6} or {x1, x2, x5, x6}. If NH1(y) = {x1, x2, x5, x6}, then x2 is adja-cent to every vertex of the 6–cycle (yx1x5x4x3x6). If NH1(y) = {x1, x2, x3, x6},then again x2 is adjacent to every vertex of 6–cycle (yx1x5x4x3x6). In anycase, W7 is produced as a subgraph of G, hence |NH1(y)| ≤ 3. Clearly, ifNH1(y) = {x1, x2, x3}, then this bound is achieved, and we conclude thatε(y,H1) ≤ 3 for all y ∈ V (B3), and hence

ε(G) ≤ ε(H1) + ε(B3) + ε(B3,H1)

≤ 13 + ⌊(n− 6)2

4⌋+ ⌊n− 5

2⌋+ 1 + 3(n− 6)

≤ ⌊n2 − 12n+ 36

4⌋+ ⌊n+ 1

2⌋ − 3 + 1 + 3n− 18

≤ ⌊n2

4⌋ − 3n+ 9 + ⌊n+ 1

2⌋ − 3 + 14 + 3n− 18

≤ ⌊n2

4⌋+ ⌊n+ 1

2⌋+ 2, for alln.

x1

x2

x3

x4

x5

x6

x7

Figure 1:

16 AHMAD AL-RHAYYEL

x1

x2

x3

x4

x5

x6

x7

Figure 2:

x1

x2

x3

x4

x5

x6

x7

Figure 3:

x1

x2

x3

x4

x5

x6

x7

x8

Figure 4:

x1

x2

x3

x4

x5

x6

x7

x8

Figure 5:


x1

x2

x3

x4

x5

x6

x7

Figure 6:

x1

x2

x3

x4

x5

x6

x7

x8

Figure 7:

x1

x2

x3

x4

x5

x6

x7

x8

x9

Figure 8:

x1

x2

x3

x4

x5

x6

x9

x7

x8

Figure 9:

18 AHMAD AL-RHAYYEL

H1 H2

Figure 10:

P Q

»»

n + 1

2n

2

Figure 11:


P Q

»

»

»

»

Figure 12:

x1

x2

x3

x4

x5 x6

Figure 13:

x1

x2

x3

x4

x5

x6

x9

x7

x8

Figure 14:

20 AHMAD AL-RHAYYEL

References

[1] A. A. Al-Rhayyel, A.M.M. Jaradat, M.M.M. Jaradat and M.S.A. Bataineh,Edge maximal graphs without WK graphs, K = 5, 6, Journal of combina-torics and number Theory, 3 (2012), 143-150.

[2] H.S. Abushahma, Edge maximal graphs without some specific graphs, M.Sc. thesis, Yarmouk University, Jordan, 2013.

[3] M.S.A. Bataineh, Some extremal problems in graph theory, Ph.D. Thesis,Curtin University of Technology, Australia, 2007.

[4] M.S.A. Bataineh and M.M.M. Jaradat, Edge maximal C2K+1-edge maximalfree graphs, Discussion Mathematicae graph theory, 32 (2012), 271-278.

[5] M.S.A. Bataineh, M.M.M. Jaradat and I.Y.A. AL-Shboul, Edge maximalgraphs without θ7-graphs, SUT Journal of Mathematics, 47 (2013), 91-103.

[6] J. Bondy, U. Murty, Graph theory with applications, MacMillan Press, Lon-don, 1976.

[7] J.A. Bondy, Large cycle in graphs, Discrete Mathematics, 1 (1971), 121-132.

[8] B. Bollobas, Extremal Graph theory, Academic press, London, 1978.

[9] L. Caccetta, K. vijayan, Maximal cycles in graphs, Discrete Mathematics,98 (1991), 1-7.

[10] G. Chartrand, P. Zhang, Introduction to graph theory, Mc Graw Hill, NewYork, 2005.

[11] P. Erdös, Extremal problems in graph theory, in M. Fidler, ed., Theory ofgraphs and its applications, Academic Press, New York (1965), 29-36.

[12] R. Häggkvist, R.J. Faudree, R.H. Schelp,Pancylic graphs-connected Ramseynumber, Ars Combin., 11 (1981), 37-49.

[13] M.M.M. Jaradat, M.S.A. Bataineh, E.Y.A. AL-Shboul, Edge–MaximalGraphs without θ2K+1 Graphs, AKCE Internaional Journal of graphs andCombinatorics, 11 (2014), 57-65.

[14] R. Jia, Some extremal problems in graph theory, Ph. D. thesis, Curtin Uni-versity of Technology, Australia, 1998.

[15] J.W. Moon, On extremal graph containing no wheels , Acta MathematicaHungerica, 16 (1965), 34-39.

[16] P. Tura’n, On an extremal problem in graph theory (in Hungarian), Math.Fiz. Lapok, 48 (1941), 436–452.


[17] G. E. Turner III, A generalization of Dirac’s theorem: Subdivision ofWheels, Discrete Mathmatics, 297 (2005), 202-205.

Accepted: 9.11.2017


On Lie ideals of inverse semirings

S. Sara∗

House no. 638, Block FGulshan Ravi, LahorePakistanPostal code 54000saro [email protected]

M. Aslam

Abstract. The purpose of this paper is to study Lie ideals of inverse semirings,thereby extending a few well-known results of I.N Herstein and C.Lanski in the settingof inverse semirings.

Keywords: inverse semiring, Lie ideals, commutators, prime inverse semiring.

1. Introduction

Lie ring structure of an associative ring R was first considered by I.N Hersteinin 1950’s. He studied Lie ideals and their relationship with R in series of papers([9],[10],[11]). In ensuing years, the study of Lie ideals generalized to prime andsemiprime rings and rings with involution. Herstein’s work on Lie ideals provedto be a source of motivation for many researchers to look into the Lie theoryin various other directions and settings ([1],[3],[4],[5],[8],[11],[14],[18],[19],[20]).Recently, M.A Javed and M.Aslam [16] introduced the notion of Lie ideals ininverse semirings and investigated some commutativity conditions on inversesemirings with the help of Lie ideals and derivations.

Our objective is to explore and generalize Lie type results of rings in thesetting of semirings, thereby we extend a few results on Lie ideals of [10],[14]and [18] to inverse semirings. These results will be helpful in investigating thetheory of Lie derivation and higher derivation of Lie ideals of semirings.

Herstein [10] proved a result on Lie ideals which says that; A non-zero Lieideal U of a 2-torsion free simple ring R is either contained in center of Ror contains a non-zero ideal of R. Later, in dealing with some problems onvon Neumann algebra, Herstein [14] extended this result to semiprime ringsas follows; Let R be a semiprime 2-torsion free ring and V be subring of R.Suppose U is a Lie ideal of R such that [V,U ] ⊂ V then either [V,U ] = 0 orV contains non-zero ideal of R. Lanski [18] proved the same result with primerings (Theorem 12 of [18]). In this paper, we establish these two main resultsof Lie ideals for prime inverse semirings.

∗. Corresponding author

ON LIE IDEALS OF INVERSE SEMIRINGS 23

By S, we mean a semiring with commutative addition and an absorbing zero.A semiring S is called an inverse semiring[17] if for every a ∈ S there exists aunique element á ∈ S such that a+ á+ a = a and á+ a+ á = á. Karvellas [17]proved that for all a, b ∈ S, (a.b)́ = á.b = a.b́ and áb́ = ab. Bandlet and Petrich[2] studied inverse semirings with some conditions (A1)-(A4). Throughout thispaper, S will denote inverse semiring which satisfies (A-2) condition of [2] i.e;for every a ∈ S, a + á ∈ Z(S), where Z(S) is center of S. This class of inversesemiring is known as MA s

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