Abstract
The power graph of a semigroup S is a simple graph with vertex set S and two distinct vertices x and y are adjacent if and only if xm = y or ym = x for some positive integer m. If m is a fixed positive integer, say k, then it is called k-power graph of S and it is denoted by . In this paper we study the structures of the components of for the multiplicative semigroup . We also study connectedness, cycle structures and symmetry of for a cyclic group Hn of order n. Finally applying some of these results we find structures and conditions for symmetries of k-power graphs of dihedral and dicyclic groups.
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