Note on the power graph of finite simple groups

چکیده مقاله

A graph is said to be 2 connected if does not have a cut vertex. The power graph P(G) of a group G is the graph which has the group elements as vertex set and two elements are adjacent if one is a power of the other. In an earlier paper, it is conjectured that there is no non-abelian finite simple group with a 2 connected power graph. Bubboloni et al. [3] and independently Doostabadi and Farrokhi D. G. [11], presented counterexamples for this conjecture. The aim of this paper is to first modify this conjecture and then prove this modified conjecture for the sporadic groups, Ree groups 2F4(q) and 2G2(q), the Chevalley groups A1(q);B2(q);C3(q) and F4(q), the unitary group U3(q), the symplectic group S4(q) and the projective special linear group PSL(3; q), where q is a prime power.