On stable and unstable trees

A graph is called stable if the adjacency matrix of the graph is nonsingular and unstable otherwise. Such terminology is related to applications in chemistry. We consider weighted graphs, this generalisation is also justified from the point of view of applications in chemistry, since it removes some restrictions in the Huckel model. Stability and instability of a weighted tree do not depend on replacements of any non-zero weights by arbitrary non-zero weights, that is, under replacement of ones in the adjacency matrix of a tree by arbitrary non-zero numbers singularity or non-singularity survives. We suggest a characterisation of stable and unstable trees which is based on a construction of trees from the so-called elementary trees.