In "Spectra of graphs" on p. 33, they state
If $\Gamma$ is primitive (strongly connected, and such that not all cycles have a length that is amultiple of some integer $d>1$), then...
Is the stated cycle condition the same as being aperiodic? If so, then why is it implied by primitivity? I know that irreducible and aperiodic imply primitive, but is the converse true as well?
For completeness, a matrix $M$ with nonegative entries is primitive if there exists a $k$ such that all entries of $M^k$ are positive.