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A. Iranmanesh,A. R. Ashrafi 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1-2
Let X be a n-set and let A = [aij] be a n×n matrix for which aij X, for 1 i, j n. A is called a generalized Latin square on X, if the following conditions is satisfied: Sni=1 aij = X = Snj =1 aij . In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a Hv−structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of order n, has a transversal set.
Iranmanesh, A.,Ashrafi, A.R. 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1
Let X be a n-set and let A = [aij] be a $n {\times} n$ matrix for which $aij {\subseteq} X$, for $1 {\le} i,\;j {\le} n$. A is called a generalized Latin square on X, if the following conditions is satisfied: $U^n_{i=1}\;aij = X = U^n_{j=1}\;aij$. In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a Hv-structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of order n, has a transversal set.
GENERATING PAIRS FOR THE SPORADIC GROUP Ru
Darafsheh, M.R.,Ashrafi, A.R. 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.12 No.1
A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group T(l, m, n) = 〈$\chi$, y, z│$\chi$$\^$l/ = y$\^$m/ = z$^n$ = $\chi$yz = 1〉. In [19], the question of finding all triples (l, m, n) such that non-abelian finite simple group are (l, m, n)-generated was posed. In this paper we partially answer this question for the sporadic group Ru. In fact, we prove that if p, q and r are prime divisors of │Ru│, where p < q < r and$.$(p, q) $\neq$ (2, 3), then Ru is (p, q, r)-generated.
REMARKS ON THE INNER POWER OF GRAPHS
JAFARI, S.,ASHRAFI, A.R.,FATH-TABAR, G.H.,TAVAKOLI, Mostafa The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.1
Let G be a graph and k is a positive integer. Hammack and Livesay in [The inner power of a graph, Ars Math. Contemp., 3 (2010), no. 2, 193-199] introduced a new graph operation $G^{(k)}$, called the $k^{th}$ inner power of G. In this paper, it is proved that if G is bipartite then $G^{(2)}$ has exactly three components such that one of them is bipartite and two others are isomorphic. As a consequence the edge frustration index of $G^{(2)}$ is computed based on the same values as for the original graph G. We also compute the first and second Zagreb indices and coindices of $G^{(2)}$.
REMARKS ON THE INNER POWER OF GRAPHS
S. JAFARI,A.R. ASHRAFI,G.H. FATH-TABAR,M. TAVAKOLI 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.1
Let G be a graph and k is a positive integer. Hammack and Livesay in [The inner power of a graph, Ars Math. Contemp., 3 (2010), no. 2, 193{199] introduced a new graph operation G(k), called the kth inner power of G. In this paper, it is proved that if G is bipartite then G(2) has exactly three components such that one of them is bipartite and two others are isomorphic. As a consequence the edge frustration index of G(2) is computed based on the same values as for the original graph G. We also compute the rst and second Zagreb indices and coindices of G(2).
The automorphism group of commuting graph of a finite group
Mahsa Mirzargar,Peter P. Pach,A. R. Ashrafi 대한수학회 2014 대한수학회보 Vol.51 No.4
Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x ≠ y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by △(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(△(G)) is abelian if and only if |G| ≤ 2 |Aut(△(G))| is of prime power if and only if |G| ≤ 2, and |Aut(△(G))| is square-free if and only if |G| ≤ 3. Some new graphs that are useful in studying the automorphism group of △(G) are presented and their main properties are investigated.
THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP
Mirzargar, Mahsa,Pach, Peter P.,Ashrafi, A.R. Korean Mathematical Society 2014 대한수학회보 Vol.51 No.4
Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and $x,y{\in}X$ ($x{\neq}y$) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ${\Delta}(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(${\Delta}(G)$) is abelian if and only if ${\mid}G{\mid}{\leq}2$; ${\mid}Aut({\Delta}(G)){\mid}$ is of prime power if and only if ${\mid}G{\mid}{\leq}2$, and ${\mid}Aut({\Delta}(G)){\mid}$ is square-free if and only if ${\mid}G{\mid}{\leq}3$. Some new graphs that are useful in studying the automorphism group of ${\Delta}(G)$ are presented and their main properties are investigated.
Szeged index of some nanotubes
H. Yousefi-Azari,B. Manoochehrian,A.R. Ashrafi 한국물리학회 2008 Current Applied Physics Vol.8 No.6
The Szeged index is one of the most important topological indices defined in chemistry. In this paper, the Szeged index of the hexagonal triangle graph T(n) and the zig-zag polyhex nanotube TUHC6[2p,q] are computed. The Szeged index is one of the most important topological indices defined in chemistry. In this paper, the Szeged index of the hexagonal triangle graph T(n) and the zig-zag polyhex nanotube TUHC6[2p,q] are computed.