We can define a representation of the automorphism group $H$ of a $n$-vertex graph $\Gamma$ as the map $\rho : H \to M$ where $M$ is the set of all $n \times n$ binary matrices.
What is the implication from the perspective of the representation if the adjacency matrices of the graph and it's automorphisms are reducible i.e. the graph is not strongly connected?
Example: Permutation cycle graphs are not strongly connected.
Permutation Cycle Graph: A permutation cycle graph for a given permutation $\pi$ of a finite set $V$, is its cycle graph $\Gamma$ such that the vertex set is $V$ and the edges are pairs $\left(\mu, \omega\right)$ for which $\pi(\mu)=\omega$.