The hh-vector of coned graphs

The coned graph Gˆ on a finite graph GG is obtained by joining each vertex of GG to a new vertex pp with a simple edge. In this work we show a combinatorial interpretation of each term in the hh-vector of Gˆ in terms of partially edge-rooted forests in the base graph GG. In particular, our interpretation does not require edge ordering. For an application, we will derive an exponential generating function for the sequence of hh-polynomials for the complete graphs. We will also give a new proof for the number of spanning trees of the wheels.