Bipartite and Planar Power Graphs of Finite Groups

The undirected power graph a(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a, b is an element of S are adjacent if and only if a not equal b and a(m) = b or b(m) = a for some positive integer m. In this paper we characterize the class of finite groups S for which g(S) is bipartite. As a consequence we prove that g(u(n)) is bipartite if and only if n = 2(k)3(l); 0 <= k <= 3, 0 <= 1 <= 1 and n >= 3. Also we prove that the power graph of the multiplicative semigroup Z(n) is bipartite if and only if n is one of 2, 3, 4,6, 8,12. We study a class of cyclic groups G for which g(G) is planar.