Defensive alliances in regular graphs and circulant graphs
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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.