### فهرست مطالب • Volume:6 Issue:2, 2019
• تاریخ انتشار: 1398/02/11
• تعداد عناوین: 9
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• Nazi Abachi, Shervin Sahebi * Pages 1-7
Let \$A\$ be a commutative ring with nonzero identity, and \$1leq n<infty\$ be an integer, and \$R=Atimes Atimescdotstimes A\$ (\$n\$ times). The total dot product graph of \$R\$ is the (undirected) graph \$TD(R)\$ with vertices \$R^*=Rsetminus {(0,0,dots,0)}\$, and two distinct vertices \$x\$ and \$y\$ are adjacent if and only if \$xcdot y=0in A\$ (where \$xcdot y\$ denote the normal dot product of \$x\$ and \$y\$).  Let \$Z(R)\$ denote the set of all zero-divisors of \$R\$.  Then the zero-divisor dot product graph of \$R\$ is the induced subgraph \$ZD(R)\$ of \$TD(R)\$ with vertices  \$Z(R)^*=Z(R)setminus {(0,0,dots,0)}\$. It follows that if  \$Gamma(A)\$ is not  perfect, then  \$ZD(R)\$ (and hence \$TD(R)\$) is not  perfect.In this paper we investigate perfectness of the graphs \$TD(R)\$ and \$ZD(R)\$.
Keywords: annihilator graph, Zero-divisor, Complete graph
• Sajjad Mahmood Robati * Pages 9-12
‎Let \$G\$ be a finite group‎. ‎We say that an element \$g\$ in \$G\$ is a vanishing element if there exists some irreducible character \$chi\$ of \$G\$ such that \$chi(g)=0\$‎. ‎In this paper‎, ‎we classify groups whose set of vanishing elements is exactly a conjugacy class‎.
Keywords: ‎Finite groups‎, ‎vanishing elements‎, ‎conjugacy classes
• Ali Reza Moniri Hamzekolaee * Pages 13-20
In this work, we introduce \$H^*\$-condition on the set of submodules of a module. Let \$M\$ be a module. We say \$M\$ satisfies \$H^*\$ provided that for every submodule \$N\$ of \$M\$, there is a direct summand\$D\$ of \$M\$ such that \$(N+D)/N\$ and \$(N+D)/D\$ are cosingular. We show that over a right perfect right \$GV\$-ring,a homomorphic image of a \$H^*\$ duo module satisfies \$H^*\$.
Keywords: \$H\$-supplemented module, cosingular module, \$H^*\$-module
Suppose \$X\$ is a simple graph. The \$X-\$join \$Gamma\$ of a set ofcomplete or empty graphs \${X_x }_{x in V(X)}\$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The \$X-\$join graph \$Gamma\$ is said to be reduced if  \$x, y in V(X)\$, \$x ne y\$ and \$N_X(x) setminus { y} = N_X(y) setminus { x}\$ imply that \$(i)\$ if \$xy notin E(X)\$ then the graphs \$X_x\$ or \$X_y\$ are non-empty; \$(ii)\$ if \$xy in E(X)\$ then \$X_x\$ or \$X_y\$ are not complete graphs. The aim of this paper is to explore how the graph theoretical properties of  \$X-\$join of graphs effect on its automorphism group. Among other results we compute the automorphism group of reduced complete-empty \$X-\$join of graphs.
Keywords: \$X-\$join of graphs, reduced \$X-\$join of graphs, automorphism group
• Fatemeh Afshari *, Mohammad Maghasedi Pages 39-45
‎Let \$Gamma\$ be a graph with adjacency eigenvalues \$lambda_1leqlambda_2leqldotsleqlambda_n\$‎. ‎Then the energy of‎ ‎\$Gamma\$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as \$mathcal{E}(G)=sum_{i=1}^n|lambda_i|\$‎. ‎Also‎ ‎the Estrada index of \$Gamma\$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as \$EE(Gamma)=sum_{i=1}^ne^{lambda_i}\$‎. ‎In this paper‎, ‎we compute the eigenvalues‎, ‎energy and Estrada index of Cayley graphs on generalized dihedral groups‎. ‎As an application‎, ‎we‎ ‎compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups‎.
Keywords: ‎Eigenvalue‎, ‎Energy of graph‎, ‎Estrada index‎, ‎Cayley graph‎, ‎Semi-Cayley graph
• Faranak Farshadifar * Pages 47-55
Let \$R\$ be a commutative ring and let \$I\$ be an ideal of \$R\$. In this paper, we will introduce the notions of 2-absorbing \$I\$-prime and 2-absorbing \$I\$-second submodules of an \$R\$-module \$M\$ as a generalization of 2-absorbing and strongly 2-absorbing second submodules of \$M\$ and explore some basic properties of these classes of modules.
Keywords: 2-absorbing submodule, weakly 2-absorbing submodule, 2-absorbing \$I\$-prime submodule, 2-absorbing second submodule, 2-absorbing \$I\$-second submodule
• Danial Khoshnevis, Zohreh Mostaghim * Pages 57-65
Let \$G\$ be a finite group. The graph \$D(G)\$ is a divisibility graph of \$G\$. Its vertex set is the non-central conjugacy class sizes of \$G\$ and there is an edge between vertices \$a\$ and \$b\$ if and only if \$a|b\$ or \$b|a\$. In this paper, we investigate the structure of the divisibility graph \$D(G)\$ for a non-solvable group with \$sigma^{ast}(G)=2\$, a finite simple group \$G\$ that satisfies the one-prime power hypothesis, a group of type(\$A\$),(\$B\$) or (\$C\$) and certain metacyclic \$p-\$groups and a minimal non-metacyclic \$p-\$group where \$p\$ is a prime number. We will show that the divisibility graph \$D(G)\$ for all of them has no triangles.
Keywords: conjugacy class, divisibility graph, metacyclic
• Masoud Zolfaghari *, Mohammad Hosein Moslemi Koopaei Pages 67-80
Let \$R\$ be a commutative ring with identity and \$M\$ be a unitary \$R\$-module. Suppose that \$phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace\$ be a function where \$S(M)\$ is the set of all submodules of \$M\$. A proper submodule \$N\$ of \$M\$ is called an \$(n-1, n)\$-\$phi\$-classical prime submodule, if whenever \$r_{1},ldots,r_{n-1}in R\$ and \$min M\$ with \$r_{1}ldots r_{n-1}min Nsetminusphi(N)\$, then \$r_{1}ldots r_{i-1}r_{i+1}ldots r_{n-1}min N\$, for some \$iinlbrace 1,ldots, n-1rbrace\$ \$(ngeqslant 3)\$.In this work, \$(n-1, n)\$-\$phi\$-classical prime submodules are studied and some results are established.
Keywords: \$psi\$-prime ideal, \$phi\$-prime submodule, n)\$-\$psi\$-prime ideal, \$(n-1, n)\$-\$phi\$-prime submodule, \$phi\$-classical prime submodule
• Zahra Heidarian * Pages 81-86
Let \$R\$ be a commutative Noetherian ring. We prove that  over a local ring \$R\$ every finitely generated \$R\$-module \$M\$ of finite Gorenstein projective dimension has a Gorenstein projective cover\$varphi:C rightarrow M\$ such that \$C\$ is finitely generated and the projective dimension of \$Kervarphi\$ is finite and \$varphi\$ is surjective.
Keywords: Cover, Precover, Gorenstein projective, totally reflexive