### فهرست مطالب

• Volume:5 Issue:3, 2016
• تاریخ انتشار: 1395/02/01
• تعداد عناوین: 5
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• Ali Reza Ashrafi*, Bijan Soleimani Pages 1-8
Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number, Sci. China Math. {\bf 56} (1) (2013) 213− − 219.] classified the connected normal edge transitive and frac12− frac12− arc-transitive Cayley graph of groups of order 4p 4p . In this paper we continue this work by classifying the connected Cayley graph of groups of order 2pq 2pq , p>q p>q are primes. As a consequence it is proved that Cay(G,S) Cay(G,S) is a frac12− frac12− edge-transitive Cayley graph of order 2pq 2pq , p>q p>q if and only if |S| |S| is an even integer greater than 2, S=TcupT −1 S=TcupT−1 and Tsubseteqcba i |0leqileqp−1 Tsubseteqcbai|0leqileqp−1 such that T T and T −1 T−1 are orbits of Aut(G,S) Aut(G,S) and begin{eqnarray*} G &=& langle a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r rangle, G &=& langle a, b, c | a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r rangle, end{eqnarray*} where r q equiv1(modp) rqequiv1(modp).
Keywords: Cayley graph, normal edge, transitive, normal arc, transitive
• David Ward* Pages 9-35
For a symmetric group G:=Sym(n) G:=Sym(n) and a conjugacy class X X of involutions in G G , it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements a,x∈X a,x∈X , either angbraca,x angbraca,x is isomorphic to the dihedral group D 8 D8 , or there is a further element y∈X y∈X such that angbraca,y≅angbracx,y≅D angbraca,y≅angbracx,y≅D8 (P. Rowley and D. Ward, On pi pi -Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). One natural generalisation of this to p p -elements is to consider when two conjugate p p -elements generate a wreath product of two cyclic groups of order p p . In this paper we give necessary and sufficient conditions for this in the case that our p p -elements have full support. These conditions relate to given matrices that are of circulant or permutation type, and corresponding polynomials that represent these matrices. We also consider the case that the elements do not have full support, and see why generalising our results to such elements would not be a natural generalisation.
Keywords: circulant matrix, cyclic group, wreath product