  In this paper we investigate the reachability and observability properties of
a network system, running a Laplacian based average consensus algorithm, when
the communication graph is a path or a cycle. More in detail, we provide
necessary and sufficient conditions, based on simple algebraic rules from
number theory, to characterize all and only the nodes from which the network
system is reachable (respectively observable). Interesting immediate
corollaries of our results are: (i) a path graph is reachable (observable) from
any single node if and only if the number of nodes of the graph is a power of
two, $n=2^i, i\in \natural$, and (ii) a cycle is reachable (observable) from
any pair of nodes if and only if $n$ is a prime number. For any set of control
(observation) nodes, we provide a closed form expression for the (unreachable)
unobservable eigenvalues and for the eigenvectors of the (unreachable)
unobservable subsystem.
