Counting unbranched subgraphs (Q1283455)

For a given graph \(G=(V,E)\), its unbranched subgraph is defined to be an edge-induced spanning subgraph such that the degree of each vertex is not greater than 2. The main result shows that the real part of any root of the polynomial \[ Q(z)=\sum_{F}z^{| F| }, \] where \(F\) is taken over all the subsets of \(E\) such that \(F\) forms an unbranched subgraph of \(G\), is always negative.