An Application of a Graph Labelling to Root Systems

The main result: If the vertices of a connected graph are labelled by positive real numbers such that the number assigned to any vertex is half of the sum of the numbers assigned to the vertices of its neighbourhood, then each label is an integral multiple of the minimum of all labels. Using this, a result proved earlier in [7] is derived: If \(V\) is a linearly dependent subset of a root system in which all roots have the same norm, then one of the roots in \(V\) is an integral combination of the other roots in \(V\).