A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. The "only if" direction is a consequence of the Perron–Frobenius theorem.
a regUlAR gRaPh of degReE K iS coNNectEd if anD oNLy iF the eIGenValue k HaS multiPLicity One. thE "oNLY if" DirEctIon iS a conSEQuence Of The pErrOn–FrObeNiUS tHeorem.