The diameter of proper power graphs of alternating groups

摘要

The power graph P(G) of finite group G is a simple graph whose vertex set is G and two distinct elements alpha and beta are adjacent if and only if one of them is a power of the other. The proper power graph of G denoted by P*(G) is a graph which is obtained by deleting the identity vertex (the identity element of G). In this paper, we improve the diameter bound of P* (A(n)) for which P*(A(n)) is connected. We show that 6 <= diam(P* (A(n))) <= 11 for n >= 51. We also describe a number of short paths in these power graphs.