2014
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THE ORDER GRAPHS OF GROUPS
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Let $G$ be a group. The order graph of $G$ is the (undirected)graph $Gamma(G)$,those whose vertices are nontrivial subgroups of $G$ and two distinctvertices $H$ and $K$ are adjacent if and only if either$o(H)o(K)$ or $o(K)o(H)$. In this paper, we investigate theinterplay between the grouptheoretic properties of $G$ and thegraphtheoretic properties of $Gamma(G)$. For a finite group$G$, we show that $Gamma(G)$ is a connected graph with diameter at mosttwo, and $Gamma(G)$ is a complete graph ifand only if $G$ is a $p$group for some prime number $p$. Furthermore,it is shown that $Gamma(G)=K_5$ if and only if either$Gcong C_{p^5}, C_3times C_3$, $C_2timesC_4$ or $Gcong Q_8$.
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10


SH.
Payrovi
Imam Khomeini International University, Qazvin  IRAN.
Imam Khomeini International University, Qazvin
Iran


H.
Pasebani
Imam Khomeini International University, Qazvin, IRAN.
Imam Khomeini International University, Qazvin,
Iran
Finite group
Connected graph
star graph
[[1] Y. Berkovich and J. Zvonimir, Group of Prime power Order, Walter de Gruyter Co. KG Berlin, New York, (2011).##[2] N. Bigss, Algebraic Graph Theory, Cambridge University Press, Cambridge, (1993).##[3] D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition, John Wiley and Sons, Inc., Hoboken, NJ, (2004).##[4] D. Gorenstein, Finite Groups, Chelsea Publishing Co. Harper, New York, (1980).##[5] I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92. American Mathematical Society, Provi##dence, RI, (2008).##[6] T. W. Hungerford, Algebra, SpringerVerlag, New York, Heidelberg and Berlin, (1989).##[7] H. E. Rose, A Course on Finite Groups , Cambridge University press, Cambridge, (1978).##[8] J. Rose, A Course on Group Theory , Cambridge University press, Cambridge, (1998).##]
ENLARGED FUNDAMENTALLY VERY THIN HvSTRUCTURES
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We study a new class of $H_v$structures called Fundamentally Very Thin. This is an extension of the well known class of the Very Thin hyperstructures. We present applications of these hyperstructures.
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T.
Vougiouklis
Democritus University of Thrace,
Democritus University of Thrace,
Greece
Hyperstructures
$H_{v}$structures
hopes
$partial$hopes
[[1] R. Bayon, N. Lygeros,Advanced results in enumeration of hyperstructures, J. Algebra, 320 (2008) 821835.##[2] P. Corsini, V. Leoreanu, Application of Hyperstructure Theory, Kluwer Academic Pub., 2003.##[3] B. Davvaz, A brief survey of the theory of Hvstructures, 8th AHA, Greece, Spanidis (2003), 3970.##[4] B. Davvaz, V. Leoreanu, Hyperring Theory and Applications, International Academic Press, 2007.##[5] B. Davvaz, R.M Santilli and T. Vougiouklis, Studies of multivalued hyperstructures for the characterization of matter##antimatter systems and their extension, Algebras, Groups and Geometries 28 (3) (2011) 05{116.##[6] C. Gutan, Proprietes des semigroupes, Applicationes a l'etude des hyperstructures tres nes, These Docteur dUni##versite, Universite Blaise Pascal, 1994.##[7] T. Vougiouklis, On representations of algebraic multivalued structures, Rivista Mat. Pura Appl., N.7, 1990, 8792.##[8] T. Vougiouklis The fundamental relation in hyperrings. The general hypereld, 4thAHA, Xanthi 1990, World Scientic##(1991), 203211.##[9] T. Vougiouklis, The very thin hypergroups and the Sconstruction, Combinatorics 88, Incidence Geometries Comb.##Str., 2, 1991, 471477.##[10] T. Vougiouklis,Hyperstructures and their Representations, Monographs in Mathematics, Hadronic, 1994.##[11] T. Vougiouklis, Enlarging Hvstructures, Algebras and Combinatorics, ICAC97, Hong Kong, Springer Verlag, 1999,##[12] T. Vougiouklis, On Hvrings and Hvrepresentations, Discrete Mathematics, Elsevier, 208/209 (1999), 615620.##[13] T. Vougiouklis, @operations and Hvelds, Acta Mathematica Sinica, English S., vol. 23, 6 (2008), 965972.##[14] T. Vougiouklis, The Liehyperalgebras and their fundamental relations, Southeast Asian Bulletin of Mathematics,##V.37(4), 2013, 601614.##[15] T. Vougiouklis, P. K.ambakiVougioukli, On the use of the bar, China USA Business Review, Vol.10, No. 6, 2011,##]
HYPERTRANSCENDENTAL FORMAL POWER SERIES OVER FIELDS OF POSITIVE CHARACTERISTIC
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2
Let $K$ be a field of characteristic$p>0$, $K[[x]]$, the ring of formal power series over $ K$,$K((x))$, the quotient field of $ K[[x]]$, and $ K(x)$ the fieldof rational functions over $K$. We shall give somecharacterizations of an algebraic function $fin K((x))$ over $K$.Let $L$ be a field of characteristic zero. The power series $finL[[x]]$ is called differentially algebraic, if it satisfies adifferential equation of the form $P(x, y, y',...)=0$, where $P$is a nontrivial polynomial. This notion is defined over fields ofcharacteristic zero and is not so significant over fields ofcharacteristic $p>0$, since $f^{(p)}=0$. We shall define ananalogue of the concept of a differentially algebraic power seriesover $K$ and we shall find some more related results.
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Habib
Sharif
Shiraz University
Shiraz University
Iran
sharif@susc.ac.ir
Formal Power Series
Algebraic Formal Power Series
Differentially Algebraic Formal Power Series
[[1] G. Christol, T. Kamae, M. MendesFrance, and G. Rauzy, Suites algebriques, automates et##substitutions, Bull. Soc. Math. France 108 (1980)401419.##[2]J. Denef and L. Lipshitz, Algebraic power series and diagonals, J. Number Theory 26 (1987)##[3] H. Furstenberg, Algebraic functions over finite fields, J. Algebra 7 (1967)271277.##[4] N. Koblitz, padic analysis; a short course on recent work, Cambridge U. P.; LMS Lecture Notes##Series 46, 1980.##[5] L. Lipshitz, The diagonal of a Dfinite power series is Dfinite, J. Algebra 113 (1988)373378.##[6] L. Lipshitz and L. Rubel, A gap theorem for power series solutions of algebraic differential##equations, Amer. J. Math., 108 (1986) 11931214. ##[7] M. MendesFrance and A. J. van der Poorten, Automata and the arithmetic of formal power##series, Acta Arith 46 (1986)211214.##[8] H. Sharif, Algebraic functions, differentially algebraic power series and Hadamard operations,##Ph.D. Thesis, Kent, 1989.##[9] —, Algebraic independence of certain formal power series (I), J. Sci. I. R. Iran, 2 (1991)5055.##[10] —, Algebraic independence of certain formal power series (II), J. Sci. I. R. Iran, 3 (1992)148##[11]—, Children products of formal power series, Math. Japonica, 38 (1993)319324.##[12] —, E algebraic functions over fields of positive characteristic an analogue of differentially##algebraic functions, J. Algebra 207 (1998)355366.##[13] —, Hadamard products of certain power series, Acta Arith. XCI (1999)95105.##[14]—and C. F.Woodcock, Algebraic functions over a field of positive characteristic and Hadamard##products, J. London Math. Soc., 37 (1988) 395403.##[15] K. ShikishimaTsuji and M. Katsura, Hypertranscendental elements of a formal power series##ring of positive characteristic, Nagoya Math. J., 125 (1992)93103.##[16] J.Y. Yao, Some transcendental functions over function fields with positive characteristic, C.##R. Acad. Sci. Paris, Series I, 334 (2002)939943.##[17] O. Zariski and P. Samuel, Commutative Algebra Vol. I, Van Nostrand, New York, 1958.##]
STABILIZER TOPOLOGY OF HOOPS
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2
In this paper, we introduce the concepts of right, left and product stabilizers on hoops and study some properties and the relation between them. And we try to find that how they can be equal and investigate that under what condition they can be filter, implicative filter, fantastic and positive implicative filter. Also, we prove that right and product stabilizers are filters and if they are proper, then they are prime filters. Then by using the right stabilizers produce a basis for a topology on hoops. We show that the generated topology by this basis is Baire, connected, locally connected and separable and we investigate the other properties of this topology. Also, by the similar way, we introduce the right, left and product stabilizers on quotient hoops and introduce the quotient topology that is generated by them and investigate that under what condition this topology is Hausdorff space, $T_{0}$ or $T_{1}$ spaces.
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R.A.
Borzooei
Shahid Beheshti University
Shahid Beheshti University
Iran


M.
Aaly Kologani
Payamenour University, Tehran
Payamenour University, Tehran
Iran
Hoop algebra
stabilizer topology
Baire space
connected
locally connected
separable topology
[[1] P. Aglian´o, I. M. A. Ferreirim, F. Montagna, Basic hoops: an algebraic study of continuous tnorm, draft, (2000).##[2] B. Bosbach, Komplement¨are Halbgruppen. Axiomatik und Arithmetik, Fundamenta Mathematicae, Vol. 64 (1969),##[3] B. Bosbach, Komplement¨are Halbgruppen. Kongruenzen and Quotienten, Fundamenta Mathematicae, Vol. 69##(1970), 114.##[4] M. Botur, A. Dvureˇcenskij, T. Kowalski, On normalvalued basic pseudohoop, Soft Comput, Vol. 16, (2012), 635##[5] N. Bourbaki, Topologie G´en´erale, Springer Berlin Heidelberg, (2007).##[6] J. R. B¨uchi, T. M. Owens, Complemented monoids and hoops, unpublished manuscript, (1975).##[7] G. Georgescu, L. Leustean, V. Preoteasa, Pseudohoops, Journal of MultipleValued logic and Soft Computing, Vol.##11. No 12, (2005), 153184.##[8] P. H´ajek, Metamathematics of fuzzy logic, Springer, Vol. 4. (1998).##[9] K. D. Joshi, Introduction to general topology, New Age International Publisher, India, (1983).##[10] Y. B. Jun, H. S. Kim, Uniform structures in positive implication algebras, Intern. Math. J. Vol. 2, No 2, (2002),##[11] Y. B. Jun, E. H. Roh, On uniformities of BCKalgebras, Commun. Korean Math. Soc. Vol. 10, No 1, (1995), 1114.##[12] M. Kondo, Some types of filters in hoops, MultipleValued Logic (ISMVL), (2011), 41st IEEE International Symposium##on. IEEE, 5053.##[13] J. R. Munkres, Topology a first course, PrenticeHall, (1975). ##[14] B. T. Sims, Fundamentals of Topology, Macmillan Publishing Co., Inc., New York, (1976).##[15] D. S. Yoon, H. S. Kim, Uniform structures in BCIalgebras, Commun. Korean Math. Soc. Vol. 17, No 3, (2002),##]
AUTOMORPHISM GROUP OF GROUPS OF ORDER pqr
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H"{o}lder in 1893 characterized all groups of order $pqr$ where $p>q>r$ are prime numbers. In this paper, by using new presentations of these groups, we compute their full automorphism group.
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M.
Ghorbani
Shahid Rajaee Teacher Training University
Shahid Rajaee Teacher Training University
Iran


F.
Nowroozi Larki
Shahid Rajaee Teacher Training University
Shahid Rajaee Teacher Training University
Iran
Affine group
Frobenius group
Automorphism group
[[1] J. E. Adney, T. Yen, Automorphisms of a pgroup, Illinois J. Math. 9 (1965) 137143.##[2] B. E. Earn ley, On nite groups whose group of automorphisms is abelian, PhD thesis, Wayne State University, Detroit, Michigan, 1975. [3] H. Christopher, R. Darren, Automorphisms of nite abelian groups, Amer. Math. Month. 114(10) (2007) 917923.##[4] M. J. Curran, Semidirect product groups with abelian automorphism groups, J. Austral. Math. Soc. Ser. A 42 (1987) 8491.##[5] D. Dummit, David, S. Foote, M. Richard, Abstract Algebra (3rd ed.), John Wiley, Sons, 2004.##[6] H. Holder, Die Gruppen der Ordnungen p3; pq2; pqr; p4, Math. Ann. xliii (1893) 371410.##[7] T. W. Hungerford , Algebra, SpringerVerlag, New York, 1980.##[8] V. K. Jain, P. K. Rai, M. K. Yadav, On Finite pgroups with abelian automorphism group, Inter. J. Alg. Compu. in press.##[9] A. Jamali, Some new nonabelian 2groups with abelian automorphism groups, J. Group Theory 5 (2002) 5357.##[10] D. Jonah, M. Konvisser, Some nonabelian pgroups with abelian automorphism groups, Arch. Math. (Basel) 26(1975) 131133.##[11] E. I. Khukhro, V. D. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small##rank, J. Algebra 301 (2006) 474492.##[12] G. A. Miller, A nonabelian group whose group of automorphisms is abelian, Messenger Math. 43 (1913) 124125.##[13] M. Morigi, On pgroups with abelian automorphism group, Rend. Sem. Mat. Univ. Padova 92 (1994) 4758.##[14] A. Ranum, The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group, Trans. Amer. Math. Soc. 8 (1907) 71{91.##[15] J. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964) 259267.##[16] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.##]
COSPECTRALITY MEASURES OF GRAPHS WITH AT MOST SIX VERTICES
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2
Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually, the origin of this concept came back to Richard Brualdi's problems that are proposed in cite{braldi}: Let $G_n$ and $G'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2 geq cdots geq lambda'_n,$$ respectively. Define the distance between the spectra of $G_n$ and $G'_n$ as$$lambda(G_n,G'_n) =sum_{i=1}^n (lambda_ilambda'_i)^2 ;;; big(text{or use}; sum_{i=1}^nlambda_ilambda'_ibig).$$Define the cospectrality of $G_n$ by$text{cs}(G_n) = min{lambda(G_n,G'_n) ;:; G'_n ;;text{not isomorphic to} ; G_n}.$Let $text{cs}_n = max{text{cs}(G_n) ;:; G_n ;;text{a graph on}; n ;text{vertices}}.$Investigation of $text{cs}(G_n)$ for special classes of graphs and finding a good upper bound on $text{cs}_n$ are two main questions in thissubject.In this paper, we briefly give some important results in this direction and then we collect all cospectrality measures of graphs with at most six vertices with respect to three norms. Also, we give the shape of all graphs that are closest (with respect to cospectrality measure) to a given graph $G$.
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67


A.
Abdollahi
University of Isfahan
University of Isfahan
Iran


Sh.
Janbaz
University of Isfahan
University of Isfahan
Iran


M.R.
Oboudi
Shiraz University
Shiraz University
Iran
Spectra of graphs
edge deletion
adjacency matrix of a graph
[[1] D. Stevanivi´c, Research problems from the Aveiro workshop on graph spectra, Linear Algebra and its Applications, 423 (2007) 172181.##[2] A. Abdollahi and M. R. Oboudi, Cospectrality of graphs, Linear Algebra and its Applications, 451 (2014) 169181.##[3] A. Abdollahi, Sh. Janbaz and M. R. Oboudi, Distance between spectra of graphs, Linear Algebra and its Applications, 466 (2015) 401408.##[4] T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, American Mathematical Society, Volume 132, 2012.##]