Boundary defensive kk-alliances in graphs

We define a boundary defensive kk-alliance in a graph as a set SS of vertices with the property that every vertex in SS has exactly kk more neighbors in SS than it has outside of SS. In this paper we study mathematical properties of boundary defensive kk-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive kk-alliance. Moreover, we consider the case in which the vertex set of a graph can be partitioned into boundary alliances, showing that if a dd-regular graph GG of order nn can be partitioned into two boundary defensive kk-alliances XX and YY, then |X|=|Y|=n2 and the algebraic connectivity of GG is equal to d−kd−k.