# Elastic energy of networks

The paper is about invariants of networks by labeling and relabeling its nodes or arcs. These invariants can be used to classify networks as to their complexity, or to distinguish non-isomorphic networks.

The basic idea of the present paper is to label the nodes of a network by (natural) umbers, which are used to compute a quantity that can be interpreted as elastic energy: the arcs are viewed as a kind of springs. By considering all possible permutations of the labels of a network one obtains a distribution of the elastic energies associated with this network. The contribution of each arc is the difference of the associated numbers with its endpoints. The distribution of the elastic energy of a network depends on its topology and is a topological invariant for this network. Computing the elastic energy for networks typically encountered in practice is impossible. The number of permutations of the nodes is then astronomical. One can only hope to make estimates of the elastic energy.