Nondiscriminatory propagation on trees

We consider a discrete-time dynamical process on graphs, firstly introduced in connection with a protocol for controlling large networks of spin 1/2 quantum mechanical particles [Burgarth and Giovannetti 2007 Phys. Rev. Lett. 99, 100501]. A description is as follows: each vertex of an initially selected set has a packet of information (the same for every element of the set), which will be distributed among vertices of the graph; a vertex v can pass its packet to an adjacent vertex w only if w is its only neighbor without the information. By means of examples, we describe some general properties, mainly concerning homeomorphism and redundant edges. We prove that the cardinality of the smallest sets propagating the information in all vertices of a balanced m-ary tree of depth k is exactly (m(k+1) + (-1)(k))/(m + 1). For binary trees, this number is related to alternating sign matrices.