Interior Symmetries and Multiple Eigenvalues for Homogeneous Networks

We analyze the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated to homogeneous networks. We consider also the special cases of regular and uniform networks. It follows from our results that the interior symmetries, as well as the reverse interior symmetries and quotient interior symmetries, of the network force the existence of eigenvalues with algebraic multiplicity greater than one. The proofs are based on the special formof the adjacencymatrices of the networks induced by those interior symmetries.