ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper CONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS CONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS Kazemi Ramin Imam Khomeini International University 01 11 2014 1 2 77 84 12 07 2014 25 12 2014 Copyright © 2014, Yazd University. 2014 http://as.yazd.ac.ir/article_493.html

The notion of a \$D\$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of  random variables on theK^{o}pka's \$D\$-Poset and prove the basic properties ofconditional expectation on this  structure.

Kopka's \$D\$-posets random variables conditional expectation
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper INDEPENDENT SETS OF SOME GRAPHS ASSOCIATED TO COMMUTATIVE RINGS INDEPENDENT SETS OF SOME GRAPHS ASSOCIATED TO COMMUTATIVE RINGS Alikhani Saeid Yazd University Mirvakili Saeed Payame Noor University 25 11 2014 1 2 85 103 25 07 2014 29 12 2014 Copyright © 2014, Yazd University. 2014 http://as.yazd.ac.ir/article_494.html

Let \$G=(V,E)\$ be a simple graph. A set \$Ssubseteq V\$ isindependent set of \$G\$,  if no two vertices of \$S\$ are adjacent.The  independence number \$alpha(G)\$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph \$Gamma(R)\$ and ideal-based zero-divisor graph \$Gamma_I(R)\$of a commutative ring \$R\$.

Independent set Independence number Zero-divisor graph, Ideal
 S. Akbari, H.R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003) 169-180.  S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004) 847-855.  S. Alikhani and Y.H. Peng, Independence roots and independence fractals of certain graphs, J. Appl. Math. Com- puting, vol. 36, no. 1-2, (2011) 89-100.  D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, The zero-divisor graph of a commutative ring, II, in: Lecture Notes in Pure and Appl. Math., vol. 220, Dekker, New York, (2001) 61-72.  D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434 447.  M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.  M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra. 6 (2005) 2043-2050.  I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208-226.  I. Gutman, F. Harary, Generalization of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106.  C. Hoede, and X. Li, Clique polynomials and independent set polynomials of graphs, Discrete Math. 25 (1994) 219-228.  N. Jafari Rad, S. H. Jafari, D.A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull. DOI:10.4153/CMB-2011-156-1 (2012).  I. Kaplansky, Commutative Rings, Chicago-London: The University of Chicago Press, 1974.  D. A. Mojdeh and A. M. Rahimi, Dominating Sets of Some Graphs Associated to Commutative Rings, Comm. Alg., 40:9 (2012) 3389-3396.  S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings 1 (4) (2002) 203-211.  S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307 (2007) 1155–1166.  S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Alg. 31, (2003) 4425-4443.  D.B. West, Introduction to Graph Theory, 2nd ed. USA: Prentice Hal, (2001).
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 unavailable ON THE SZEGED INDEX OF NON-COMMUTATIVE GRAPH OF GENERAL LINEAR GROUP ON THE SZEGED INDEX OF NON-COMMUTATIVE GRAPH OF GENERAL LINEAR GROUP Azad Azizollah Arak University Elahinezhad Nafiseh Arak University 20 11 2014 1 2 105 115 20 06 2014 29 12 2014 Copyright © 2014, Yazd University. 2014 http://as.yazd.ac.ir/article_495.html

Let \$G\$ be a non-abelian group and let \$Z(G)\$ be the center of \$G\$. Associate with \$G\$ there is agraph \$Gamma_G\$ as follows: Take \$Gsetminus Z(G)\$ as vertices of\$Gamma_G\$ and joint two distinct vertices \$x\$ and \$y\$ whenever\$yxneq yx\$. \$Gamma_G\$ is called the non-commuting graph of \$G\$. In recent years many interesting works have been done in non-commutative graph of groups. Computing the clique number, chromatic number, Szeged index and  Wiener index play important role in graph theory. In particular, the clique number of non-commuting graph of some the general linear groups has been determined. nt Recently, Wiener and Szeged indiceshave been computed for \$Gamma_{PSL(2,q)}\$, where \$qequiv 0 (mod~~4)\$. In this paper we will compute the Szeged index for\$Gamma_{PSL(2,q)}\$, where \$qnotequiv 0 (mod ~~ 4)\$.

Non-commuting grapg general Linear group Szeged index
 A. Abdollahi, S. Akbari, and H.R. Maimani, Non-commuting graph of a group, J. Algbera 298, 468-492, (2006).  A. Azad, M. Eliasi, Distance in the non-commuting graph of groups, ARS Combin. 99, 279-287, (2011).  A. Azad, Mohammad A. Iranmanesh, Cheryl E. Praeger and P. Spiga, Abelian coverings of finite general linear groups and an application to their non-commuting graphs, J. Algebr. Comb. 34, 638-710, (2011).  A. Azad, Cheryl E. Praeger, Maximal set of pairwise non-commuting elements of three-dimensional general linear groups, Bull. Aust. Math. Soc. 0, 1-14, (2009).  F. Buckley, and F. Harary, F. Distance in graphs, Addison-Wesley, Redwood, CA, 1990.  D.M. Cvetkocic, M. Doob, H. Sachs, Spectar of graph theory and application, Academic Press, New York, 1980.  P. Dankelmann, Average distance and independence numbers, Discrete Appl. Math., 51, 75-83, (1994).  A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math., 66, 211-249, (2001).  A.A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publications De L‘Institut math´ematiques Nouvelle s´erie, 56, 18-22, (1994).  I. Gutman, A formula for the Wiener number of trees and its extension to graphs cycles, Graph theory Notes of New York, 27, 9-15, (1994).  B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.  B. Huppert, and N. Blackburn, Finite groups III, Springer-Verlag, Berlin, 1982.  S. Klavzˇar, J. Jerebic, and D.F. Rall, Distance-balanced graphs, Anna. Comb., 12, 71-79, (2008).  S. Klavzˇar, A. Ajapakes, and I. Gutman, The Szeged and Wiener index of graphs, Appl. Math. Lett., 9 (5), 45-49,  M. Mirzargar, and A.R. Ashrafi, Some distance-based topologigcal indices of a non-commuting graph, Hacettepe J.Math. Stat. 41(4), 515-526, (2012).  A.R. Moghaddamfar, W.J. Shi, W. Zhou and A.R. Zokayi, On the noncommuting graph associated with a finite group, Siberian Math. J. 46 (2), 325-332, (2005).
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper A SHORT NOTE ON ATOMS AND COATOMS IN SUBGROUP LATTICES OF GROUPS A SHORT NOTE ON ATOMS AND COATOMS IN SUBGROUP LATTICES OF GROUPS Khass Hossain University of Kashan Bazigaran Behnam University of Kashan Ashrafi Ali Reza University of Kashan 01 03 2015 1 2 117 122 25 09 2014 29 12 2014 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_518.html

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.

Atom Coatom group Lattice
 H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, N.J., 1956.  L. Fuchs, Infinite abelian groups (Hungarian), Mat. Lapok (N.S.) 11 (2002/03), no. 1, 16–26 (2006).  A. Y. Ol’shanskii, Geometry of Defining Relations in Groups, Translated from the 1989 Russian original by Yu. A.Bakhturin, Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991.  P. P. P´alfy, Groups and lattices. Groups St. Andrews 2001 in Oxford, Vol. II, 428–454, London Math. Soc. Lecture Note Ser., 305, Cambridge Univ. Press, Cambridge, 2003.  Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth edition, Graduate Texts in Mathematics, 148. Springer-Verlag, New York, 1995.  M. Suzuki, Structure of a group and the structure of its lattice of subgroups, Ergebnisse der Mathematik und ihrerGrenzgebiete, Neue Folge, Heft 10. Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1956.
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper NILPOTENT GRAPHS OF MATRIX ALGEBRAS NILPOTENT GRAPHS OF MATRIX ALGEBRAS Mahmoodi A. Payame Noor University 20 02 2015 1 2 123 132 12 12 2014 08 05 2015 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_519.html

Let \$R\$ be a ring with unity. The undirected nilpotent graph of \$R\$, denoted by \$Gamma_N(R)\$, is a graph with vertex set ~\$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}\$, and two distinct vertices \$x\$ and \$y\$ are adjacent if and only if \$xy in N(R)\$, or equivalently, \$yx in N(R)\$, where \$N(R)\$ denoted the nilpotent elements of \$R\$. Recently, it has been proved that if \$R\$ is a left Artinian ring, then \$diam(Gamma_{N}(R))leqslant 3\$. In this paper, using the concept of rank over commutative rings, we investigate basic properties of undirected nilpotent graph of matrix algebras. Moreover, some result on undirected nilpotent graph of matrix algebras over commutative rings are given. For instance, we prove that \$Gamma_{N}(M_{n}(R))\$ is not planar for all \$ngeqslant 2\$. Furthermore, we show that \$diam(Gamma_{N}(R))leqslant diam(Gamma_{N}(M_{n}(R)))\$ for an Artinian commutative ring \$R\$. Also, we prove that \$Gamma_{N}(M_{n}(R))congGamma_{N}(M_{n}(T(R)))\$, where \$T(R)\$ be the total quotient ring of a commutative ring \$R\$

zero-divisor graph Nilpotent Graph Commutative Ring
 S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270, 169-180 (2003).  S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296, 462{479 (2006).  D. F. Anderson, A. Frazier, A. Lauve, P. Livingston, The zero-divisor graph of a commutative ring. II Lect. Notes Pure and Appl. Math 220, 61{72 (2001).  D. F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, Von-Neumann regular rings and Boolean algebras, J. Pure Appl. Algebra 180, 221{241 (2003).  D. F. Anderson and P. Livingston. The zero-divisor graph of a commutative ring, J. Algebra 217, 434{447 (1999).  D. F. Anderson, S. B. Mulay,On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210, 543-550(2007).  I. Beck, Coloring of commutative rings. J. Algebra 116, 208{226 (1988).  B. Fine, Classi cation of nite rings of order p2, Mathematics Magazine, VOL. 66, NO. 4, 246{252 (1993)  J.A. Bondy, U.S.R. Murty,Graph Theory with Applications, American Elsevier, New York, 1976.  I. Bozic, Z. Petrovic,Zero-divisor graphs of matrices over commutative rings, Communication in Algebra 37, 1186{ 1192 (2009)  W. C. Brown, Matrices Over Commutative Rings, Marcel Dekker, Inc, New York, Basel, Hong Kong.  P. W. Chen, A kind of graph structure of rings, Algebra Colloq 10(2), 229{238 (2003).  T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, Inc, 1976.  A. H. Li,Q. H. Li, A kind of graph structure on non-reduced rings, Algebra Colloq 17(1), 173{180 (2010).  A. H. Li, Q. S. Li, A kind of graph structure on Von-Neumann regular rings, International J. Algebra 4, 291-302 (2010).
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper SIGNLESS LAPLACIAN SPECTRAL MOMENTS OF GRAPHS AND ORDERING SOME GRAPHS WITH RESPECT TO THEM SIGNLESS LAPLACIAN SPECTRAL MOMENTS OF GRAPHS Taghvaee Fatemeh University of Kashan Fath-Tabar Gholam Hossein University of Kashan 31 12 2014 1 2 133 141 10 09 2014 30 12 2014 Copyright © 2014, Yazd University. 2014 http://as.yazd.ac.ir/article_520.html

Let \$G = (V, E)\$ be a simple graph. Denote by \$D(G)\$ the diagonal matrix \$diag(d_1,cdots,d_n)\$, where \$d_i\$ is the degree of vertex \$i\$  and  \$A(G)\$ the adjacency matrix of \$G\$. The  signless Laplacianmatrix of \$G\$ is \$Q(G) = D(G) + A(G)\$ and the \$k-\$th signless Laplacian spectral moment of  graph \$G\$ is defined as \$T_k(G)=sum_{i=1}^{n}q_i^{k}\$, \$kgeqslant 0\$, where \$q_1\$,\$q_2\$, \$cdots\$, \$q_n\$ are the eigenvalues of the signless Laplacian matrix of \$G\$. In this paper we first compute  the \$k-\$th signless Laplacian  spectral moments of a graph for small \$k\$  and then we order some graphs with respect to the signless Laplacian  spectral moments.

Spectral moments sequence signless Laplacian generalized Petersen graph T−order
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