The -vector of coned graphs

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