In this paper we study defensive alliances in some regular graphs.
We determine which subgraphs could a critical defensive alliance
of a graph $G$ induce, if $G$ is $6$-regular and the cardinality of the alliance is
at most $8$.
In particular, we study the case of circulant graphs, i.e. Cayley graphs on a cyclic group.
The critical defensive alliances of a circulant graph of degree at most $6$
are completely determined. For the general case, we give tight lower and upper bounds
on the alliance number of a circulant graph with $d$ generators.