Nondiscriminatory Propagation on Trees

TL;DR

This paper studies a discrete-time process on graphs related to quantum network control, characterizing minimal initial sets needed for full information propagation on trees, with exact formulas and connections to combinatorics.

Contribution

It provides a precise formula for the smallest propagating sets on balanced m-ary trees and explores properties like homeomorphism and redundant edges.

Findings

Exact formula for minimal propagating sets on balanced m-ary trees.

Connection between binary tree propagating sets and alternating sign matrices.

Insights into properties like homeomorphism and redundant edges in the process.

Abstract

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 [Phys. Rev. Lett. 99, 100501 (2007)]. 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 neighbour without the information. By mean 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.